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^  TREATISE  ON  PHYSIOLOGY  AND  HYGIENE. 

FOR  EDUCATIONAL  INSTITUTIONS  AND  THE   GENERAL  READER. 

By  Joseph  C.  Hutchison,  M.D., 

President  of  the  New  York  Pathological  Society;  Vice-President  of  the  New  York 

Academy  of  Medicine;  Surgeon  to  the  Brooklyn  City  Hospital;  and  late 

President  of  the  Medical  Society  of  the  State  of  New  York. 


Fully  Illustrated  with  ITumerous  Elegant  Engravings.    12aio.    300  pages. 


1.  Tlie  Plan  of  tlie  Work  is  to  present  the  leading  facts  and  principles 
of  human  Physiology  and  Hygiene  in  language  so  clear  and  concise  as 
to  be  readily  comprehended  by  pupils  in  schools  and  colleges,  as  well  as 
by  general  readers  not  f  amilar  with  the  "subject.  2.  The  Style  is  terse 
and  concise,  yet  intelligible  and  clear;  and  all  useless  technicalities  have 
been  avoided.  3.  The  Bange  of  Subjects  Treated  includes  those  on  which 
it  is  believed  all  persons  should  be  informed,  and  that  are  proper  in  a 
work  of  this  class.  4.  Ihe  Subject-moMer. — The  attempt  has  been  made 
to  bring  the  subject-matter  up  to  date,  and  to  include  the  results  of  the 
most  valuable  of  recent  researches  to  the  exclusion  of  exploded  notions 
and  theories.  Neither  subject — Physiology  or  Hygiene — has  been  elabo- 
rated at  the  expense  of  the  other,  but  each  rather  has  been  accorded  its 
due  weight,  consideration,  and  space.  The  subject  of  Anatomy  is  in- 
cidentally treated  with  all  the  fullness  the  author  believes  necessary  in  a 
work  of  this  class.  5.  The  Engravings  are  numerous,  of  great  artistic 
merit,  and  are  far  superior  to  those  in  any  other  work  of  the  kind, 
among  them  being  tv>^o  elegant  colored  plates,  one  showing  the  Viscera 
in  Position,  the  other,  the  Circulation  of  the  Blood.  6.  The  Size  of  the 
work  will  commend  itself  to  teachers.  It  contains  about  800  pages,  and 
can  therefore  be  easily  completed  in  one  or  two  school  terms. 

The  publishers  are  confident  that  teachers  will  find  this  work  full  of  valuable 
matter,  much  of  which  cannot  be  found  elsewhere  in  a  class  manual,  and  so  pre- 
sented and  arranged  that  the  book  can  be  used  both  with  pleasiu-e  and  success  in 
the  schoolroom. 

"  Many  of  the  popular  works  on  Physiology  now  in  use  in  schools,  academies,  and 
colleges,  do  not  reflect  the  present  state  of  the  science,  and  some  of  them  abound 
in  absolute  errors.  The  work  which  Dr.  Hutchison  has  given  to  the  public  is  free 
from  these  objectionable  features.  I  give  it  my  hearty  commendation." — Samuel 
G.  Armor,  M.D.,  late  Professor  in  Ifichiijan  University. 

"This  book  is  one  of  the  very  few  school  books  on  these  sub jects  which  can  be 
imconditionally  recommended.  It  is  accurate,  free  from  needless  technicalities, 
and  judicious  m  the  practical  advice  it  gives  on  Hygienic  topics.  The  illustrations 
are  excellent,  and  the  book  is  well  printed  and  bound.  "—Boston  Journal  of 
Chemistry. 

"Just  the  thing  for  schools,  and  I  sincerely  hope  that  it  maybe  appreciated  for 
what  it  is  worth,  for  we  are  certainly  in  need  of  books  of  this  kind." — Prof.  Austin 
Flint,  Jr.,  Professor  of  Physiology  in  Bellevue  Hospital  Kedical  College,  JS^ew  York 
City,  and  author  of  "  Physiology  of  il/a?!,"  etc.,  etc. 

"I  have  read  it  from  preface  to  colophon,  and  find  it  a  most  desirable  text-book 
for  schools.  Its  matter  is  judiciously  selected,  lucidly  presented,  attractively 
treated,  and  pointedly  illustrated  by  memorable  facts:  and,  as  to  the  plates  and 
diagrams,  they  are  not  only  clear  and  intelUgible  to  beginners,  but  beautiful  speci- 
mens of  engraving.  I  do  not  see  that  any  better  presentation  of  the  subject  of 
physiology  could  be  given  ■v\'ithin  the  same  compass." — Prof.  John  Ordronaux, 
Professor  of  Physiology  in  the  University  of  Vermont,  and  also  in  the  2^'ational 
Medical  College,  Washington,  D.  C. 

The  above  work  is  the  viost  popular  loork  on  the  above  subjects  yet  published.     It  is 
used  in  thousands  of  schools  tvith  marked  success. 

PnbHshed  bj  CLAEK  &  MAYNABD,  New  York. 


EDUCATION  DEPT. 


Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/completegradedarOOthomrich 


THOMSON'S    ARITHMETIC^A^K\  \SERIES 

IN    TWO     BOOKSi       ,  ^    ''\'''.  '        ■     , 


COMPLETE 

GRADED 

ARITH  M  ETIC, 

Oral   and   Written, 


UPON    THE 


INDUCTIVE      METHOD      OF      INSTRUCTION^ 


FOR  SCHOOLS  AND  ACADEMIES. 


James   B.  Thomson,   LL.  D., 

AUTHOR    OF    MATHEMATICAL    SERIES. 


NEW   YORK: 

Clark   k   Maynard,    Publishers, 

734  Broadway. 

CHICAGO  :   46  Madison  Street. 

1884. 


THOMSON'S   MEW   ARITHMETICAL   SERIES 

IN     T^A^O     BOOKS.  Q  f\     10  3. 
1-^* .. 

I.    First  Lessons  in  Arithmetic, 

Oral   and   Written.     Illustrated.  ^^ 

(For  Primary  Schools.)  fo^-^^-'VoAr  . 

II.   Complete  Graded  Arithmetic, 

Oral  and   Written.      In  one  Volume. 
(For  Schools  and   Academies.) 

Key  to  Complete  Graded  Arithmetic. 

(For  Teachers.) 


THOMSON'S    MATHEMATICAL    SERIES. 

illustrated  table-book. 

NEW    rudiments    OF    ARITHMETIC. 

complete    INTELLECTUAL    ARITHMETIC. 

NEW    PRACTICAL    ARITHMETIC. 

KEY    TO    PRACTICAL    ARITHMETIC. 

HIGHER    ARITHMETIC. 

KEY    TO    HIGHER    ARITHMETIC, 
PRACTICAL    ALGEBRA. 

KEY    TO    PRACTICAL    ALGEBRA. 

COLLEGIATE    ALGEBRA. 

KEY    TO    COLLEGIATE    ALGEBRA. 

GEOMETRY  AND  TRIGONOMETRY.    (In  preparation,) 


Copyright^  1882,  by  James  B.    Thofnson. 


Smith  k  McDougal,  Electrotyper.s. 
82  BeekmaD  St.,  N.  Y. 


TT/n  *  rPTrVNT      mTOT 


■t  [^ 


E  E  F  A  O  E 


rpHE  book  now  offered  to  the  public,  unites  in  one  volume  Oral  and 
Written  Arithmetic  upon  the  inductive  method  of  instruction.  Its 
aim  is  two-fold  :  to  develop  the  intellect  of  the  pupil,  and  to  prepare  him 
for  the  actual  business  of  life.  In  securing  these  objects,  it  takes  the 
most  direct  road  to  a  practical  knowledge  of  Arithmetic. 

The  pupil  is  led  by  a  few  simple  appropriate  examples  to  infer  for  him- 
self the  general  principles  upon  which  the  operations  and  rules  depend, 
instead  of  taking  them  upon  the  authority  of  the  author  without  explana- 
tion. He  is  thus  taught  to  put  the  steps  of  particular  solutions  into  a 
concise  statement,  or  general  formula.  This  method  of  developing  princi- 
ples is  an  important  feature. 

It  has  been  a  cardinal  point  to  make  the  explanations  simple,  the  steps 
in  the  reasoning  short  and  logical,  and  the  definitions  and  rules  brief,  clear, 
and  comprehensive. 

The  discussion  of  topics  which  belong  exclusively  to  the  higher  depart- 
ments of  the  science  is  avoided ;  while  subjects  deemed  too  difficult  to  be 
appreciated  by  beginners,  but  important  for  them  when  more  advanced, 
are  placed  in  the  Appendix,  to  be  used  at  the  discretion  of  the  teacher. 

Arithmetical  puzzles  and  paradoxes,  and  problems  relating  to  subjects 
having  a  demoralizing  tendency,  as  gambling,  etc.,  are  excluded.  All  that 
is  obsolete  in  the  former  Tables  of  Weights  and  Measures  is  eliminated, 
and  the  part  retained  is  corrected  in  accordance  with  present  law  and  usage. 

Examples  for  Practice,  Problems  for  Review,  and  Test  Questions  are 
abundant  in  number  and  variety,  and  all  are  different  from  those  in  the 
Practical  Arithmetic. 

iviS4*J34B 


iv  Preface, 

The  arrangement  of  subjects  Is  systematic;  no  principle  is  anticipated, 
or  used  in  the  explanation  of  another,  until  it  has  itself  been  explained. 
Subjects  intimately  connected  are  grouped  together  in  the  order  of  their 
dependence. 

In  connection  with  the  Notation  of  Integers,  that  of  Decimals  is  taught 
to  three  places  below  units,  corresponding  to  dimes,  cents,  and  mills. 
Decimal  or  Metric  Weights  and  Measures  are  placed  next  after  Decimal 
Currency.  Percentage  is  followed  by  its  applications  in  their  proper  order, 
as  Profit  and  Loss,  Commission,  etc. 

General  Analysis,  covering  the  several  departments  of  Commercial 
Arithmetic,  has  received  special  attention.  The  articles  devoted  to  Test 
Questions,  and  to  the  Entrance  Problems  of  different  Colleges,  will  be 
found  a  valuable  addition  and  excellent  practice.  Thanks  are  due  to  the 
College  OflBcers  who  have  kindly  furnished  copies  of  their  Examination 
Papers. 

In  the  preparation  of  the  work  the  author  has  carefully  weighed  tlie 
discussions  in  the  various  journals  of  education  respecting  the  present 
wants  of  our  schools,  and  has  endeavored  to  provide  for  them.  He  has 
availed  himself  of  many  valuable  suggestions  from  business  men,  practical 
teachers,  and  educators,  all  of  whom  he  desires  to  thank  most  cordially  for 
the  aid  they  have  rendered. 

He  cheerfully  submits  the  result  of  his  labors  to  his  friends,  the  teach- 
ers, and  the  public,  for  whose  favorable  verdict  upon  his  former  efforts  he 

desires  to  express  renewed  obligations. 

J.  B.  T. 
Brooklyn,  April,  1882. 


:^Bz 


ONTENTS. 


<S^ 


PAGE 

Notation    anil   Numera- 

tion ^ 

Arabic  Notation 8 

Applied  to  Decimals 13 

Numeration  Table 13 

Addition 1^ 

Development  of  Principles.    .  19 
Adding  U.  S.  Money 21 

Subtraction 26 

Development  of  Principles. . .  27 
Subtracting  U.  S.  Money 30 

Multiplication 36 

Development  of  Principles...   38 
Multiplying  U.  S.  Money  ....  39 

Division 49 

Development  of  Principles...  52 

Dividing  by  Long  Division ...  52 

Dividing  by  Short  Division.  . .  53 

Dividing  U.  S.  Money 54 

General  Principles 60 

Cancellation 65 

Properties  of  Numbers.     67 

Development  of  Principles 68 

Factoring: 69 

Common  Divisors 71 

Common  Multiples. 74 

Development  of  Principles.  . .   74 

Common  Fractions 77 

General  Principles 79 

Reduction  of  Fractions .  .•  80 

Common  Denominators 83 

Addition  of  Fractions 85 

Subtraction         "         88 

Multiplication    "         90 


PAGE 

General  Rule 95 

Division  of  Fractions 96 

General  Rule 101 

Reducing  Complex  Fractions.  103 
Finding    a  Number   from  a 
given  part 106 

Decimal  Fractions HI 

Notation  of  Decimals 112 

Reduction  of  Decimals 114 

Addition  of  Decimals 118 

Subtraction         "         119 

Multiplication    "         120 

Division  "         122 

Development  of  Principles..   122 

Decimal  Currency 125 

U.  S.  Money 126 

Short  Methods 129 

Accounts  and  Bills 1 33 

Metric  System 137 

Add,    Subtract,    etc.,  Metric 
Numbers 146 

Compound  Nuniber§. ...  149 

Reduction 161 

Denominate  Fractions 164 

Changing  Metric  to  Common 

Weights  and  Measures.  . . .  166 
Compound  Addition 167 

"  Subtraction 169 

Exact  time  between  two  dates  170 
Compound  Multiplication...    172 

"     '       Division 172 

Longitude  and  Time 173 

Measurement  of  Surfaces 176 

Solids 179 

Board  Measure 182 

Rectangular  Cisterns,  etc 184 


VI 


Contents. 


PAGE 

Percentage 189 

Finding  the  Percentage 193 

"        Amt.  or  Diflerence. .  194 

Rate. 196 

Base 198 

Profit  and  Loss 200 

Commission  and  Brokerage . .  204 

Insurance 206 

Taxes 209 

Duties  or  Customs 211 

Interest 213 

General  Method 215 

Six  per  cent  Method 217 

Method  by  Days 219 

Partial  Payts.,  U.  S.  Rule. . .   220 

Mercantile  Method 223 

Problems  in  Interest, 224 

Compound  Interest. 226 

Discount 229 

True  Discount 229 

Bank        "       230 

Commercial  Discount 232 

Equation  of  Payments . .  234 

Development  of  Principles. . .  234 

Averaging  Accounts 238 

Stocks 241 

Finding  the  Premium,  etc.  . .   242 

Excliang-e 248 

Domestic  Exchange 249 

Foreign  "        251 

Table  of  Money  Units 252 

General  Analysis 255 

Ratio 271 


PAGE 

Proportion 274 

Cause  and  Effect 279 

Compound  Proportion 281 

Partitive  "  284 

Partnerslii]> 286 

Involution 289 

Evolution 292 

Square  Root 293 

Right-angled  Triangles   ....  299 

Cube  Root 303 

Illustration  by  Cu.  Blocks. . .  305 

Progression    311 

Arithmetical  Progression. . . .  311 
Geometrical  "  316 

Mensuration    319 

Triangles 320 

Quadrilaterals 323 

Circles 326 

Solids 328 

Gauging  of  Casks 333 

Tonnage  of  Vessels 334 

Test  Questions 335 

College  Entrance  Papers. . . .  353 

Appendix 359 

Roman  Notation 359 

Contractions 361 

Demonstration  of  g.  c.  d.  . .   363 

Circulating  Decimals 365 

Surveyors'  Measure 368 

Government  Lands 369 

Apothecaries'  Fluid  Measure.   372 

Annual  Interest 373 

Partial  Payts.  Ct.,  Vt.,  N.  H.  374 
12%  Method  of  Casting  Int. .  378 
Average  of  Mixtures 378 


QJ  <  " 


RITHMETIC. 


Definitions. 

Art.  1.   A  Unit  is  one  or  any  single  thing;  as,   one,  one 
book,  one  chair. 

2.  A  Number  is  a  imif  or  a  collection  of  units. 

Thus,  one,  or  one  book,  is  a  unit ;   five,  or  five  books,  is  a  collection 
of  units. 

3.  The  Unit  of  a  Number  is  one  of  the  collection  forming 
that  number. 

Thus,  the  unit  of  four  books  is  one  book,  of  seven  is  one. 

4.  An  Abstract  Number  is  one  that  is  7iot  applied  to  any 
object. 

Thus,  four,  five,  thirteen,  etc.,  are  abstract  numbers. 

5.  A  Concrete  Number  is  one  that  is  applied  to  some  object. 
Thus,  five  boys,  seven  apples,  etc.,  are  concrete  numbers. 

6.  Like   Numbers  are   those    which   express  units  of  the 
same  ki7id. 

Thus,  eight  pears  and  five  pears,  four  and  seven,  are  like  numbers. 

7.  Unlike    Numbers    are    those    which    express    units    of 
different  kinds. 

Thus,  seven,  six  peaches,  nine  days,  are  unlike  numbers. 

8.  Numbers  are  expressed  by  words,  by  figures,  or  by  letters. 

9.  Arithmetic  is  the  science  which  treats  of  numhers  and 
their  applications. 


r^ 


•>>..  o 


i     \ -<  ♦ 


OTATIO]^  AND  NUMERATION. 


10.  Notation  is  expressing  numbers  by  figures  or  letters. 

11.  Numeration  is  reading  numbers  expressed  by  figures 
or  letters. 

Note. — There  are  two  methods  of  Notation,  called  the  Arabic  and  the 
Romnn.  The  former  is  the  method  in  general  use,  and  is  so  called  be- 
cause it  was  introduced  into  Europe  in  the  loth  century  by  the  Arabians. 

12.  The  Arabic  Notation  expresses  numbers  by  teyi 
different  characters,  called  Figures ;  viz., 

/,   2,   3,    J,,   5,    6,    7,    8,    (j,    0. 

One,    Two,     Tliree,  Four,  Five,      Six,      Seven,  Figlit,  Nine,  Naught. 

13.  The  first  nine  are  called  Significant  figures,  because 
they  always  express  some  value.     They  are  also  called  Digits. 

14.  The  last  one  is  called  Naught,  because  when  standing 
alone  it  has  no  value.     It  is  also  called  Cipher  or  Zero. 

15.  The  Value  of  a  figure  is  the  numher  it  represents. 

16.  Nine  is  the  largest  number  expressed  by  one  figure. 

17.  The  significant  figures  standing  alone,  express  single 
things  or  ones ;  as,  4  apples,  5,  7. 

18.  To  express  the  numbers  from  7ii7ie  to  one  liundred 
requires  two  figures  written  side  by  side. 

19.  The  first  figure  at  the  right  denotes  Ones,  wdiich  are 
called  Units  of  the  First  Order. 

20.  The  figure  in  the  second  place  denotes  ten  ones,  which 
are  called  Tens,  or  Units  of  the  Second  Order. 

Thus,  the  figures  35,  denote  5  ones  and  3  tens,  and  are  read,  "Thirty- 
five." 


Notation  and  Numeration. 
Write  the  following  numbers  in  figures  : 


1.  Twenty-five. 

5.  Thirty-six. 

2.  Tliirtj-eiglit. 

6.  Forty-nine. 

3.  Fifty-six. 

7.   Fifty-four. 

4.  Forty-two. 

8.  Sixty-eight. 

Eead  the  following 

numbers : 

13.      63.               17. 

43.            21.     64. 

14.      54.                18. 

38.            22.     57. 

15.      49.               19. 

69.            23.     76. 

16.      78.               20. 

84.            24.     92. 

9.  Seventy- three. 

10.  Fifty-nine. 

11.  Eighty-eight. 

12.  Ninety-nine. 


25.  75. 

26.  88. 

27.  93. 

28.  99. 

21.  Ninety-nine  is   the  largest  number  which    can   be   ex- 
pressed by  two  figures. 

22.  To  express  the  numbers  from  ninety-nine  to  07ie  thou- 
sand, requires  tliree  figures  written  side  by  side. 

23.  The  figure  in  the  third  place  denotes  ten  tens,  which 
are  called  Hundreds,  or  Units  of  the  Third  Order. 

Thus,  the  figures  436,  denote  4  hundred,  3  tens,  and  6  units,  and  are 
read,  "  Four  hundred  thirty-six." 

Write  the  following  numbers  in  figures  : 

29.  Two  hundred  forty-six.  33.  Five  hundred  eight. 

30.  Three  hundred  fifty-four.  34.   Six  hundred  seventy. 

31.  Five  hundred  thirty-two.  35.  Eight  Iiundred  three. 

32.  Four  hundred  fifty.  36.  K'ine  hundred  ninety-nine. 

Read  the  following  numbers  :* 

37.  243.  41.  632.  45. 

38.  420.  42.  567.  46. 

39.  364.  43.  740.  47. 

40.  419.  44.  321.  48. 

24.  Nine  Hundred  Ninety-nine  is   the  largest  number  ex- 
pressed by  three  figures. 

*  In   reading  numbers  expressed  by  three  or   more  figures,   omit  the  word  and 
after  hundreds. 


407. 

49. 

830. 

536. 

50. 

604. 

249. 

51. 

783. 

680. 

52. 

999. 

10  Notation  and  Numeration, 

25.  To  express  larger  numbers,  other  orders  of  units  are 
formed,  called  thousands,  ten-thousands,  hundred-thousands, 
millmis,  etc. 

26.  A  figure  in  the  fourth  place  denotes  Thousands,  which 
are  called  Units  of  the  Fourth  Order. 

27c  A  figure  in  the  fifth  place  denotes  Ten-thousands,  which 
are  called  Units  of  the  Fifth  Order. 

28.  A  figure  in  the  sixth  place  denotes  Hundred-thousands, 
which  are  called  Units  of  the  Sixth  Order. 

29.  A  figure  in  the  seventh  place  denotes  Millions,  which 
are  called  Units  of  the  Seventh  Order. 

30.  If  any  orders  are  omitted,  ciphers  must  be  written  in 
their  places. 

Thus,  four  tliousand  three  hundred  five,  is  written  4305. 
The  figures  5046,  denote  5  thousands,  0  hundreds,  4  tens,  and  6  units, 
and  are  read,  "  Five  thousand  forty-six." 

Write  the  following  in  figures  : 

53.  Three  thousand  two  hundred  sixty-eight. 

54.  Five  thousand  seventy-five. 

55.  Six  thousand  three  hundred  ten. 

56.  Seven  thousand  fifty-three. 

57.  Eight  thousand  seven  hundred  five. 

58.  Nine  thousand  nine  hundred,  ninety-nine. 

Read  the  following  numbers  : 

59.  1265.  63.  3420.  67.  5101. 

60.  1503.  64.  3051.  68.  5049. 

61.  2034.  65.  4036.  69.  6008. 

62.  2105.  66.  4003.  70.  7059. 

31.  The  different  values  of  units  expressed  by  the  significant 
figures,  are  determined  by  the  place  they  occupy,  and  are 
called  simple  and  local  values. 


Notation  and  Nameration, 


11 


These  values  are  illustrated  by  the  following  diagram 


1000.  100.  10.  1. 

32.  The  Simple  Value  of  the  units  represented  by  the  sig- 
nifieaDt  figures  is  the  number  which  they  represent  when 
standing  alone  or  in  units  place. 

33.  The  Local  Value  of  these  units  is  the  number  which 
they  represent  wdien  standing  oq  either  side  of  units  place. 

Thus,  2  standing  alone,  or  in  the  jirst  place,  denotes  2  simple  units  ;  in 
the  second  place,  it  denotes  2  tens,  as  in  25  ;  in  the  third  place,  it  denotes 
2  hundreds,  as  in  246,  etc. 

Note. — These  different  orders  of  units  correspond  to  dollars,  dimes, 
and  cents.  Thus,  10  cents  make  1  dime,  10  dimes  1  dollar.  Xow  a  cent 
is  a  unit,  a  dime  is  a  unit,  and  a  dollar  is  a  unit ;  but  these  miits  have 
diiferent  values,  corresponding  to  the  orders  of  units. 

34.  From  the  above  illustrations  w^e  derive  the  following 

Principles. 

j?°.  Ten  units  of  any  order  mahe  a  unit  of  the  next 
higher  order. 

2°.  Moving  a  figitre  one  place  to  tlie  left,  increases  its  value 
ten  times. 

3°.  Moving  a  figiire  one  place  to  the  right,  diminishes  its 
value  ten  times. 

35.  Hence,  the  great  law  of  the  Arabic  Notation,  viz.: 

The  Orders  of  Units  increase  and  decrease  by  the  uniform 
scale  of  Ten. 

The  Arabic  Notation  is  therefore  called  the  Decimal  System, 
from  the  Latin  word  decern,  wiiich  means  ten^ 


12  Notation  and  Numeration, 

36.  To  Express  Decimal  Parts  of  a  Unit. 

By  the  law  of  the  decimal  notation  a  unit  of  the  tliird  order 
is  ten  times  a  unit  of  the  second  order ;  a  unit  of  the  second 
order  is  ten  times  a  simple  unit  or  one. 

By  extending  this  law  heloiu  units,  a  simple  unit  is  ten  times 
a  unit  of  the  first  decimal  order ;  a  unit  of  the  first  decimal 
order  is  ten  times  a  unit  of  the  second  decimal  order,  and  so  on. 
In  this  way  a  series  of  orders  is  formed  below  units  which  regu- 
larly decrease  by  the  scale  of  ten. 

37.  The  first  order  on  the  right  of  units  is  called  tenths ; 
the  second,  hundredths  ;  the  third,  thousandths ;  etc. 

38.  These  lower  orders  are  separated  from  units  by  a  period 
(.)  called  the  Decimal  Point. 

39.  The  orders  on  the  left  of  the  decimal  point  are  called 
Whole  Numbers  or  Integers ;  those  on  the  rigid,  Decimals. 

Thus,  seven  and  five  tenths  are  written  7.5;  nine  and  fifty-three 
hundredths  are  written  9.53  ;  sixty-five  and  two  hundred  seventy-three 
thousandths  are  written  65.273.  The  figures  4.7  denote  four  ones  and 
seven  tenths  of  one,  and  are  read,  "  Four  and  seven  tenths."  The  figures 
6.35  denote  six  ones  and  thirty-five  hundredths  of  one,  etc. 

Write  the  following  in  figures : 

1.  Eive  tenths.  7.   63  and  7  hundredths. 

2.  Four  hundredths.  8.   3  units  and  5  thousandths. 

3.  Sixty-five  hundredths.  9.  245  and  25  hundredths. 

4.  Seventeen  thousandtlis.  10.   7  and  62  hundredths. 

5.  Forty-two  thousandths.  11.  456  and  273  thousandths. 

6.  Fifty-four  thousandths.  12.  503  and  6  tliousandths. 

Bead  the  following  : 

13.  3.7.  17.     62.3.  21.     0.25.  25.       42.365. 

14.  5.24.         18.     75.21.         22.     0.7.  26.     125.034. 

15.  23.9.  19.     36.45.         23.     0.253.  27.     245.007. 

16.  31.25.         20.     68.4.  24.     0.45.  28.     360.248. 


Notation  and  Nmneration, 


13 


40.  The  French  Method  of  writing  and  reading  large  num- 
bers, is  shown  in  the  following 


Numeration     Ta  b  le 


Names  of 

Periods.        Trillions.       Billions.       Millions.    Thousands.       Units. 


Thou- 
sandths. 


~ 

a 

;-, 

Orders  of 

'^ 

. 

m 

.Q 
'i 

o 

7^ 

^ 

o 

Units, 

;-i 

i; 

_o 

'^3 

<o 

K 

&H 

Eh 

1— ( 

Number. 

6 

5 

1     , 

,  7 

K   H   pq 


W 


s        2    c    o 
§        K   H   ^ 


o      a    5 


,780,900,240,78  5 


I»      9^      33 

a  -s  § 

a  -  o 
o  s  .a 
E-i  K  H 

3  2  4 

5th  Per.    4th  Period.    3d  Period.    2d  Period.    1st  Period.       Decimals. 

41.  The  first  period  on  the  left  of  the  decimal  point 
expresses  units,  tens,  and  hundreds,  and  is  called  Units 
Period ;  the  second  period  denotes  thousands,  etc.,  and  is 
called  Thousands  Period ;  and  so  on. 

42.  Beginning  at  unit's  place,  the  orders  on  the  right  of  the 
decimal  point  express  tenths,  hundredths,  thousandths,  etc. 

The  number  in  the  table  is  read,  '^Six  hundred  fifty-one 
trillions,  seven  hundred  eighty  billions,  nine  hundred  millions, 
two  hundred  forty  thousand,  seven  hundred  eighty-five,  and 
three  hundred  twenty-four  thousandths." 

43.  To  express  larger  numbers,  oilier  periods  are  formed  in 
like  manner,  called  Trillions,  Quadrillions,  Quintillions,  Sex- 
tillions,  Septillions,  Octillions,  Nonillions,  Decillions,  etc. 


44.   To  Express  Numbers  by  Figures  : 

Begin  at  the  left  and  write  the  figures  of  the  given 
orders  in  succession  toivards  the  right.  If  any  orders 
are  omitted,  supply  their  places  by  ciphers,  and  separate 
tenths  from  units  hy  a  decimal  point. 


14  Notation  and  Numeration, 

45.   To  Read  Numbers  expressed  by  Figures: 

Separate  the  number  into  periods  of  three  figures 
each,  counting  each  ivay  from  units  place.  Begin  at 
the  left,  read  each  period  separately,  and  add  the 
name  to   each,    except  units  period. 

WheJi  there  are  decimals,  read  the  figures  on  the 
right  of  the  decimal  point  as  if  whole  numbers,  and 
add  the  name  of  the  loivest  oj^der. 

Tims,  the  figures  256,347.259  are  read,  "Two  hundred  fifty-six  tliou- 
saiid,  three  hundred  forty-seven,  ay^tZ  two  hundred  fifty-nine  thousandths." 

Notes. — 1.  In  numerating  decimals  as  well  as  whole  numbers,  the 
vnits  place  should  always  be  made  the  starting  point. 

2.  In  reading  whole  numbers  and  decimals,  the  word  and  should 
be  used  between  the  whole  number  and  the  decimals. 

Write  the  following  numbers  in  figures  : 

1.  One  thousand,  forty-two. 

2.  Thirty  thousand,  nine  hundred  seven. 

3.  Forty-six  thousand,  four  hundred. 

4.  Ninety-two  thousand,  one  hundred  eight. 

5.  Sixty-eight  thousand,  seventy. 

6.  One  hundred  twenty-four  thousand,  six  hundred. 

7.  Two  hundred  thousand,  one  hundred  sixty. 

8.  Four  hundred  five  thousand,  forty-five. 

9.  Three  hundred  forty  thousand. 

10.  Nine  hundred  thousand,  seven  hundred  twenty. 

11.  One  million,  seven  hundred  thousand. 

12.  Thirty  million,  twenty  thousand,  fifty. 

13.  Three,  and  seven  tenths. 

14.  Forty-five  hundredths. 

15.  Twenty-eight,  and  three  hundredths. 

16.  Two  hundred  fifty,  and  seven  thousandths. 

17.  Thirty-five,  and  twenty-four  thousandths. 

18.  Four  hundred  two  thousand,  and  eight  tenths. 

19.  Seventeen  hundred,  and  twenty-five  thousandths. 

20.  Nine  thousand,   two   hundred  five,   and    twenty-three 
hundredths. 


Notation  and  Numeration,  15 

Point  off,  numerate,  and  read  the  following  numbers : 


1. 

520. 

15. 

207047. 

29. 

0.23. 

2. 

603. 

16. 

2605401. 

30. 

0.06. 

3. 

4506. 

17. 

4040680. 

31. 

0.235. 

4. 

7045. 

18. 

5604700. 

32. 

0.047. 

5. 

8700. 

19. 

2020105. 

33. 

4.05. 

6. 

25008. 

20. 

45001003. 

34. 

6.078. 

7. 

40625. 

21. 

30407045. 

35. 

0.265. 

8. 

75407. 

22. 

145560800. 

36. 

8.003. 

9. 

125242. 

23. 

8900401. 

37. 

9.036. 

10. 

240251. 

24. 

250708590. 

38. 

261.54. 

11. 

407203. 

25. 

803068003. 

39. 

24.06. 

12. 

300200. 

26. 

2175240670. 

40. 

3.807. 

13. 

1255673. 

27. 

7240305060. 

41. 

20.964. 

14. 

5704086. 

28. 

0.4. 

42. 

523.604. 

[For  Rom.  Notation  and  Eng.  Numeration,  see  Arts.  860-863,  Appendix.] 

Questions. 

1.  What  is  a  unit?  2.  Number?  3.  Tlie  unit  of  a  number?  4.  An 
abstract  number?      5.  Concrete?     6.  Like  numbers?     7.  Unlike? 

9.  What  is  Arithmetic ?  10.  Notation  ?  11.  Numeration?  12.  The 
Arabic  Notation  ?  15.  What  is  the  value  of  a  figure  ?  19.  What  does  the 
first  figure  at  the  right  denote?     20.  In  the  second  place?     What  called? 

23,  In  the  third  place  ?  What  called  ?  24.  What  is  the  largest  num- 
ber expressed  by  three  figures  ? 

25.  How  are  larger  numbers  expressed?  26.  What  does  a  figure  in  the 
fourth  place  denote  ?  27.  In  the  fifth  ?  28.  In  the  sixth  ?  29.  In  the 
seventh  ?    30.  If  any  orders  are  omitted,  what  is  to  be  done  ? 

31.  What  are  the  different  values  expressed  by  figures  called?  How 
determined  ?  82.  What  is  the  simple  value  of  a  figure  ?  33.  The 
local  value  ? 

34.  Name  the  three  principles  of  Notation  ?  35.  What  is  the  great  law 
of  the  Arabic  Notation  ?  What  is  it  often  called  ?  Why  ?  36.  Explain 
how  this  law  is  applied  in  expressing  parts  of  one? 

87.  What  is  the  first  place  at  the  right  of  units  called  ?  The  second  ? 
Third  ?  38.  How  are  these  orders  separated  from  units  ?  89.  Wliat  are 
the  orders  at  the  left  called  ?     Those  at  the  right  ? 

40.  Repeat  the  Numeration  Table?  41.  What  is  the  first  period  on  the 
left  of  the  decimal  point  called  ?  The  second  ?  The  third  ?  42.  The 
first  period  on  the  right  of  units  ?  43.  How  are  larger  numbers 
expressed  ?    44.  How  express  numbers  by  figures  ?     4o.  How  read  them  ? 


DDITION. 


Oral     Exercises. 

46.  1.  How  many  are  7  marbles,  5  marbles,  and  9  marbles  ? 
Solution. — 7  and  5  are  12  and  9  are  21.  Ans.  31  marbles. 

2.  How  many  dollars  are  15  dollars,  10  dollars  and  6  dollars  ? 

3.  How  many  are  12  books,  8  books,  and  7  books  ? 

4.  How  many  upits  are  15  units,  6  units,  and  10  units  ? 

5.  How  many  are  6,  8,  and  4  ?     7,  5,  and  6  ? 

6.  If  one  tree  bears  12  peaches,  another  9,  another  6,  how 
many  peaches  will  all  bear  ? 

7.  A  dairy-woman  put  17  pounds  of  butter  into  a  stone  jar, 
and  afterwards  added  8  pounds  more  ;  how  many  pounds  of 
butter  did  the  jar  contain  ? 

8.  If  you  have  12  pens  in  your  box,  and  afterwards  add 
7  more,  how  many  pens  will  your  box  contain  ? 

9.  If  a  class  of  12  pupils  is  united  with  a  class  of  15  pupils, 
how  many  will  be  in  the  class  ? 

10.  If  9  units  and  7  units  and  8  units  are  united  in  one 
number,  how  many  units  wall  the  number  contain? 

Definitions. 

47.  Addition  is  uniting  two  or  more  numbers  in  one. 

The  ansiver  or  number  found-by  addition  is  called  the  Sum  or 
Amount. 

Note. — The  Sum  or  Amount  contains  as  many  units  as  all  the 
numbers  added. 

48.  The  Sign  of  Addition  is  +.  It  is  called  Plus,  which 
means  more,  and  shows  that  the  numbers  between  which  it  is 
placed  are  to  be  added.     It  is  read  "  and,"  or  "  added  to." 

Thus,  5  +  3  is  read,  "  5  plus  3,"  "  5  and  3,"  or  "  5  added  to  3," 


Addition.  17 

49.  The  Sign  of  Equality  is  =.  It  is  read,  '^ equal"  or 
"is  equal  to,"  and  shows  that  the  numbers  between  which  it  is 
placed  are  equal. 

Thus,  the  expression  5  +  4  =  9,  is  read,  "  5  plus  4  equal  9,"  or  "  the 
sum  of  5  and  4  is  equal  to  9." 

How  many  are 


11. 

6  +  5? 

19. 

3+2+1 ? 

27. 

7+5+6+2? 

12. 

7  +  4? 

20. 

2+4+3? 

28. 

3+6+0+8? 

13. 

5  +  3? 

21. 

5+7+2? 

29. 

4+7+5+2? 

14. 

6  +  8? 

22. 

2+4+6? 

30. 

8+3+2+6? 

15. 

7  +  7? 

23. 

6+1+8? 

31. 

3+5+9+4? 

16. 

4  +  9? 

24. 

7+0+9? 

32. 

7+6+8+5? 

17. 

8  +  3? 

25. 

4+9+2? 

33. 

8+7+9+6? 

18. 

9  +  6? 

26. 

7+6+9? 

34. 

9+8+7+4? 

35.  If  you  pay  10  cents  for  an  inkstand,  8  cents  for  paper, 
4  cents  for  pens,  how  much  will  you  pay  for  all  ? 

36.  If  you  pick  7  apples  from  one  tree,  5  from  another,  and 
6  from  another,  how  many  apples  will  you  have? 

37.  What  is  the  sum  of  9  dollars,  7  dollars,  and  4  dollars  ? 

38.  How  many  are  11  books,  10  books,  and  6  books  ? 

39.  If  you  pay  10  cents  for  fare,  15  cents  for  lunch,  and 
8  cents  for  fruit,  how  much  will  you  have  spent  ? 

Slate     Exercises. 

50.  Write  the  following  in  columns  and  add  each  up  and 
down  several  times,  naming  results  only. 

1.  Add  4,  2,  6,  3,  5,  6,  4,  5,  3,  4,  and  2. 
Thus,  4,  6,  12,  15,  30,  26,  etc. 

2.  Add  5,  3,  4,  2,  3,  5,  6,  2,  7,  4,  and  3. 

3.  Add  4,  3,  5,  7,  6,  2,  3,  7,  4,  6,  4,  and  5. 

4.  Add  2,  5,  4,  3,  7,  6,  4,  3,  2,  4,  5,  and  6. 

5.  Add  8,  2,  7,  6,  5,  3,  1,  7,  6,  8,  7,  4,  3,  and  7. 

6.  Add  9,  4,  2,  7,  3,  4,  5,  6,  8,  4,  6,  5,  4,  and  8. 

7.  Add  7,  5,  3,  6,  4,  2,  5,  3,  8,  1,  6,  3,  4,  and  9. 

8.  Add  6,  3,  4,  7,  2,  9,  4,  3,  1,  8,  5,  2,  6,  4,  and  7.. 


18  Addition. 

9.  Add  8,  4,  2,  7,  5,  3,  6,  2,  4,  6,  5,  8,  1,  9,  5,  6,  and  8. 

10.  Add  6,  8,  3,  5,  2,  7,  4,  6,  3,  1,  7,  6,  5,  4,  6,  8,  and  9. 

11.  What  is  the  sum  of  3232,  20,  4314,  and  2123  ? 

Explanation. — We  write  the  numbers  so  that  units  operation. 

of  the  same  order  stand  in  the  same  column,  and  naming  3232 

results  only,  proceed  thus,    3,  7,  9   units.     Write  the  9  oq 

under  units  column. 

Adding  the  tens  in  the  same  manner,  2,  3,  5,  8  tens. 
Write  the  8  under  tens  column,  and  add  the  hundreds  ^123 

and  thousands  in  like  manner.  Ans     9689 

We  prove  the  work  by  beginning  at  the  top  and  adding 
each  column  downward.     The  two  results  agree,  therefore  the  work  is 
right. 

Add  and  prove  the  following  : 


(12.) 

(13.) 

(14.) 

(15.) 

4121 

3204 

5202 

2320 

304 

4050 

43 

3054 

1052 

23 

430 

412 

3402 

612 

2304 

4012 

16.  A  man  paid  5423   dollars  for   a  house,  325   dollars  for 
repairs,  and  430  dollars  for  painting ;  what  did  the  whole  cost  ? 

17.  If  a  steamer  goes  243  miles  in  one  day,  321  miles  the 
next,  and  402  miles  the  third,  how  far  does  she  go  in  3  days  ?. 

18.  A  father  gave  314  acres  to  one  son,  241  acres  to  another, 
and  432  acres  to  another  ;  how  many  acres  did  he  give  to  all  ? 

Oral     Drill. 

51.     1.  Add  by  2's  from  0  to  50,  naming  results  only. 
Thus,  two,  four,  six,  eight,  ten,  etc. 

Add  in  like  manner 

2.  By  2's  from  1  to  51.  6.  By  4's  from  0  to  64. 

3.  By  3's  from  0  to  60.  7.  By  4's  from  1  to  65. 

4.  By  3's  from  1  to  61.  8.  By  4's  from  2  to  (jQ. 

5.  By  3's  from  2  to  62.  9.  By  4's  from  3  to  67. 

10.  Add  the  other  digits  5,  6,  7,  8,  9  in  the  same  mannjcr,  till 
the  result  becomes  as  large  as  may  be  deemed  desirable. 


Addition,,  19 


Development   of   JPbinciples. 

52.  1.  What  is  the  sum  of  5  peaches  and  8  peaches  ? 

2.  What  is  the  sum  of  7  apples  and  5  marbles  ? 

A?is.  Apples  and  marbles  are  unlike  numbers  and  cannot  be 
added.     (Art.  7.) 

3.  What  is  the  sum  of  7  units  and  9  units?     Of  3  tens  and 
5  tens  ? 

4.  Is  the  sum  of  4  tens  and  5  units  equal  to  9  tens  or  9 
units  ? 

Ans.  Tens  and  units  are  unliJce  orders  and  cannot  be  added 
to  each  other. 

5.  Which  is  the  greater,  the  sum  of  4  +  5  +  6,  or  of  6  +  5  +  4  ? 
Ans.  The  sums  are  equal. 

53.  From  the  above  examples  we  infer  the  following 

Principles. 

i°.   07ily  like  numbers  and  like  orders  of  units  can  be  added 
one  to  another. 

2°.   The  sum  is  the  same  in  whatever  order  numhers  may  be 
added. 

Oral     Exercises. 

54.  1.  How  many  are  40  pounds  and  50  pounds? 

Analysis. — 40  is  equal  to  4  tens  and  50  is  equal  to  5  tens ;  now  4  tens 
and  5  tens  are  9  tens  or  90.  Ans.  90  pounds. 

Note. — In  adding  large  numbers  mentally,  it  is  more  convenient  and 
expeditious  to  begin  with  the  highest  orders. 

2.  How  many  are  30  and  40  ?     50  and  60  ?     80  and  70  ? 

3.  If  I  pay  56  dollars  for  a  sleigh  and  37  dollars  for  a  cart, 
what  will  both  cost  ? 

Analysis. — 56  equals  5  tens  and  6  units,  and  37  equals  3  tens  and 
7  units.  Now  5  tens  and  3  tens  are  8  tens,  or  80,  and  6  units  and  7  units 
are  13  units,  or  1  ten  and  3  units  which  added  to  80  make  93  dollars,  Ans, 


20  Addition, 

4.  A  farmer  had  2  wheat  fields  ;  one  produced  68  bushels, 
the  other  75  bushels  ;  how  many  bushels  did  both  produce  ? 

5.  If  Charles  reads  64  pages  one  day,  and  78  pages  the  next 
day,  how  many  pages  will  he  read  in  both  days  ? 

6.  A  drover  bought  87  sheep  of  one  man  and  98  of  another; 
how  many  sheep  did  he  buy  of  both  ? 

7.  A  lad  having  spent  40  cts.  finds  he  has  37  cts.  left ;  how 
much  had  he  at  first  ? 

8.  How  many  are  78  and  97  ? 

9.  How  many  are  84  and  69  ? 

10.  A  man  divided  his  farm  into  2  parts,  one  of  which  con- 
tained 77  acres  and  the  other  88  acres  ;  how  many  acres  were 
in  his  farm  ? 

11.  What  is  the  sum  of  43  +  28  +  7  ? 

12.  What  is  the  sum  of  52  +  43  +  9  ? 

13.  The  price  of  a  horse  is  94  dollars,  of  a  buggy  is  62  dollars, 
and  a  saddle  is  1 0  dollars  ;  what  is  the  price  of  all  ? 

Written     Exercises. 
55.  When  the  Sum  of  a  Column  exceeds  9. 
1.  What  is  the  sum  of  4524,  276,  6745  and  5498  ? 

Explanation. — Since  like  orders  only  can  be  added,  operation. 

for  convenience  we  write  them  under  each  other,  and  4524 

beginning  at  the  right,  add  each  column    separately,  276 

naming  the  results.   Thus,  adding  the  first  column,  8, 13,  6745 

19,  23  units,  or  2  tens  and  3  units,  we  write  the  3  units  kaqq, 

under  the  column  of  units,  and  add  the  2  tens  to  the  

column  of  tens  because  they  are  like  orders.  Ans.    17043 

Adding  the  next  column,  2,  11,  15,  22,  34  tens,  or  2  hundreds  and  4  tens, 
we  set  the  4,  or  units  fio^ure  of  the  sum,  under  the  column  added,  and  add 
the  tens  figure  to  the  next  column. 

Again,  the  sum  of  the  next  column  is  20  hundreds,  or  2  thousands  and 
no  hundreds.  We"  place  a  cipher  under  the  column,  and  add  the  2  to  the 
next  column. 

The  sum  of  this  column  is  17  thousands,  and  being  the  last,  we  set 
down  the  whole  sum. 

Note, — The  process  of  reserving  the  tens  and  adding  them  to  the 
nest  column,  is  called  "carrying  the  tens." 


Addition,  21 

Add  and  explain  the  following  in  like  manner  : 

(2.)  (3.)                       (4.)                           (5.) 

3506  yards.  4672  rods.  7845  weeks.  8407  pounds. 

4824     "  89    ''  468      "  3400      " 

719     "  725    "  5030      "  7902      " 

3005     ''  9306    "  6404      ''  8434      " 

56.  To  Add  Decimals,  or  U.  S.  Money. 

6.  What  is  the  sum  of  235.267;  75.43;  8.624;  0.238;  and 
362.07  ? 

OPERATION. 

Explanation.— We  write  the  numbers  so  that  the  '  ^        ' 

same  orders  stand  in  the  same  column,  and  beginning  75. 4o 
with  the  lowest  add  each  column,  and  set  down  the  8.624 

result  as  before,  placing  the  decimal  point  under  those  0.238 

in  the  numbers  added.  op.n  a/v 

Ans.  681.629 

7.  Find  the  sum  of  65.7;  248.62;  40.255;  54.07;  and 
6.389. 

8.  Find  the  sum  of  3.036;  0.75;  23.008;  0.236;  and 
87.604. 

57.  The  denominations  of  TJ.  S.  Money  increase  and  decrease 
by  tens,  and  are  expressed  in  the  same  manner  as  n)liole  numbers 
and  decimals.     (Art.  39.) 

58.  Dollars  are  Integers,  and  occupy  the  place  of  whole 
numbers.  Ce7its  occupy  the  place  of  tenths  and  hundredths, 
and  mills  the  place  of  thousandths.  Dollars  are  separated  from 
cents  by  the  decimal  point.     (Art.  39.) 

b9.  The  Dollar  Sign  is  %.     Thus,  $25  is  read,  ^'25  dollars.*' 

The  Sign  for  Cents  is  (p,  or  ct. ;  as,  17^,  or  17  ct. 

The  expression  $64,735  is  read,  "  Sixty-four  dollars,  seventy-three  cents 
and  five  mills." 

Note. — As  two  places,  tenths  and  hundredths,  are  occupied  by  cents,  if 
the  number  of  cents  is  less  than  10,  a  cipher  must  be  placed  before  the 
figure  representing  them.     Thus,  seven  cents  are  written  $0.07. 


22 

Addition. 

9.  Add  $38.273 ; 

$80.46; 

$5,073 

;  and  $0.85. 

SUGGESTION.- 

—Write  dollars  under  dollars,  cents  under 

cents,  and  i 

as  above. 

Arts.  $124,656. 

(10.) 

(11.) 

(12.) 

(13.) 

$42,213 

136.23 

123.463 

$42,282 

4.30 

10.826 

14.052 

2.20 

30.034 

4.05 

40.201 

23.034 

12.42 

53.615 

23.124 

26.317 

60.  From  the  preceding  Exercises  we  infer  the  following 

General     Rule. 

/.  Write  the  numbers  so  that  units  of  the  same  order 
stand  in  the  same  column. 

II.  v4.dd  the  right  hand  column,  and  placing  the  units 
of  the  sum  under  it,  add  the  tens  to  the  next  order. 

III.  Proceed  thus  with  the  other  columns,  writing  the 
whole  sum  of  the  last.  If  there  are  decimals,  place  the 
decimal  point  of  the  sum  under  those  in  the  numbers 
added. 

Proof. — Add  the  numbers  from  the  top  downward, 
and  if  the  two  results  agree  the  worh  is  right.     (Art.  S°.) 

Ajt'PLICA.TIONS, 

1.  Four  men  formed  a  partnership ;  A  furnished  $2878, 
B  $1784,  0  $1265,  and  D  $894.  What  was  the  amount  of 
their  capital  ? 

2.  A  man  sold  three  house  lots ;  for  one  he  received  $975, 
for  another  $763,  and  for  the  third  $586.  What  did  the  whole 
amount  to  ? 

3.  A  gentleman  purchased  a  store  for  $4500,  and  paid  $75 
for  repairs,  and  $150  for  having  it  enlarged.  For  how  much 
must  he  sell  it,  in  order  to  gain  $175  ? 

4.  A  merchant  paid  $375  for  one  package  of  goods,  $467  for 
another,  $254  for  another,  and  $348  for  another.  How  much 
did  he  pay  for  all  ? 


Addition.  23 

5.  A  certain  orchard  contains  256  apple  trees,  119  peach 
trees,  83  phim  trees,  and  45  pear  trees.  How  many  trees  are 
there  in  the  orchard  ? 

6.  A  man  being  asked  his  age,  said  he  was  17  years  old  when 
he  left  the  academy,  he  spent  4  years  in  college,  3  years  in  a 
Jaw  school,  practiced  law  15  years,  was  a  member  of  congress 
18  years,  and  it  was  16  years  since  he  retired  from  business 
How  old  was  he  ? 

7.  A  shopkeeper  having  a  note  due,  paid  $184  at  one  time, 
at  another  $268,  at  another  1379,  at  another  $467,  and  there 
were  $350  still  unpaid.     What  was  the  amount  of  his  note  ? 

8.  A  gentleman  owds  a  house  worth  $10800,  a  store  worth 
$5450,  a  farm  worth  $3700,  and  has  $15000  personal  property. 
What  is  the  amount  of  his  estate  ? 

9.  A  man  left  his  estate  to  his  wife,  his  three  sons,  and  two 
daughters ;  to  his  wife  he  gave  $10350,  to  his  sons  $5450 
apiece,  and  his  daughters  $3500  apiece.  How  much  was  he 
worth  ? 


Add  and  prove  the  following : 

10.  261  +  31  -f  256  -h  17  ? 

11.  163  4-  478  +  82  +  19? 

12.  428  +  632  +  76  +  394? 

13.  320  +  856  +  100  +  503? 

14.  641  +  108  +  138  +  710? 

15.  700  +  66  +  970  +  21  ? 

16.  304  +  971  +  608  +  496  ? 

17.  848  +  683  +  420  +  668  ? 

18.  868  +  45  +  17  +  25  +  27  +  38  ? 

19.  641  +  85  +  580  +  42  +  7  +  63  ? 

20.  425  +  346  +  681  +  384  ? 

21.  135  +  342  +  778  +  528  ? 

22.  460  +  845  +  576  +  723  ? 

23.  2345  +  4088  +  260  +  819  ? 

24.  8990  +  5632  +  5863  +  756  ? 

25.  2842  +  6361  +  523  +  836  ? 

26.  602  +  173  +  586  +  408  +  973  ? 

27.  424  +  375  +  626  +  75  +  855  ? 


24  Addition. 

28.  A  man  wishing  to  stock  his  farm,  paid  1197  for  horses, 
$86  for  oxen,  $175  for  cows,  and  $169  for  sheep.  How  much 
did  he  give  for  the  whole  ? 

29.  A  butcher  sold  to  one  customer  157  pounds  of  meat,  to 
another  159,  to  another  149,  to  another  97,  and  to  another 
68  pounds.     How  much  did  he  sell  to  all  ? 

30.  A  carpenter  received  1879  for  one  job,  for  another  $786, 
for  another  $693,  for  another  $587,  for  another  $476,  and  for 
another  $368.     How  much  did  he  receive  in  all  ? 

31.  A  merchant  pays  $560  a  year  for  store  rent,  $1386  to 
one  clerk,  $1267  to  another,  and  $369  for  various  other 
expenses.     AVhat  does  his  business  cost  him  a  year? 

32.  A  man  receives  $568  rent  for  one  store,  $479  for  another, 
and  $276  for  another.    How  much  does  he  receive  for  them  all? 


(33.) 

(34.) 

(35.) 

(36.) 

$75,340 

$68,901 

$64,268 

$346,768 

6,731 

50,345 

405 

21,380 

748 

75,005 

1,708 

4,075 

68,451 

29,450 

43,671 

126,849 

396 

80,063 

72,049 

257 

7,503 

91,700 

492 

1,305 

46,075 

43,621 

1,760 

24,350 

1,290 

47,834 

25,357 

439,871 

25,738 

83,276 

1,434 

40,306 

46,803 

25,327 

84,162 

601,734 

37.  What  is  the  sum  of  five  billions,  ten  millions  forty-five ; 
eight  millions,  eight  thousand,  eight;  two  billions,  four  hun- 
dred thirty  millions,  two  hundred  thousand,  four  hundred  ? 

38.  A  man  paid  $2243  for  a  house,  $825  for  a  barn,  and  for 
his  farm  as  much  as  for  his  house  and  barn  together;  how 
much  did  he  pay  for  his  farm  ;  and  how  much  for  all  ? 

39.  A  man  having  7  children  gave  a  farm  to  each  worth 
$2378  ;  what  was  the  value  of  all  their  farms  ? 

40.  A  man  bequeathed  $6275  apiece  to  his  3  children,  and 
to  his  wife  the  balance  of  his  property,  which  was  equal  to  the 
amount  he  gave  all  his  children  ;  what  was  he  worth  ? 


Addition,  25 

41.  Sir  Isaac  Newton  was  born  in  the  year  1642,  and  died  in 
his  eighty-fifth  year ;  in  what  year  did  he  die  ? 

42.  Four  men,  A,  B,  C,  and  D,  built  a  school-house;  A  gave 
11500,  B  $1750,  0  11975,  and  D  gave  the  land,  which  was 
worth  as  much  as  A  and  B  gave  ;  what  was  the  whole  cost  ? 

The  Census  Eeport  of  1880  gives  the  population  of  the  U.  S. 
as  follows : 

43.  Maine,  648,936;  N.  H.,  346,991;  Vt.,  332,286;  Mass., 
1,783,085;  R.  L,  276,531;  Conn.,  622,700;  K  Y.,  5,082,871; 
N.  J.,  1,131,116 ;  Penn.,  4,282,891.  What  was  the  population 
of  the  North  Atlantic  States  ? 

44.  Delaware,  146,608  ;  Md.,  934,943  ;  D.  C,  177,624;  Va., 
1,512,565;  West  Va.,  618,457;  N.  C,  1,399,750;  S.  C, 
995,577;  Ga.,  1,542,180;  Fla.,  269,493.  What  was  the  popu- 
lation of  the  Souili  Atlantic  States? 

45.  Ohio,  3,198,062;  Ind.,  1,978,301 ;  111,3,077,871;  Mich., 
1,636,937;  Wis.,  1,315,497;  Minn.,  780,773;  la.,  1,624,615  ; 
Mo.,  2,168,380;  Dak.,  135,177;  Xeb.,  452,402  ;  Kan.,  996,096. 
What  was  the  population  of  the  Northern  Central  States  .^^ 

46.  Alabama,  1,262,505;  Miss.,  1,131,597;  La.,  939,946; 
Texas,  1,591,749;  Ark.,  802,525;  Temi.,  1,542,359;  Ky., 
1,648,690.  What  was  the  population  of  the  Southern  Central 
States  ? 

47.  California,  864, 694;  Col.,  194,327;  Or.,  174,768  ;  Wash., 
75,116;  Id.,  32,610;  Mon.,  39,159;  Wy.,  20,789;  Utah, 
143,963;  Arizona,  40,440;  Nev.,  62,266;  New  Mex.,  119,565. 
What  was  the  population  of  the  Western  States  and  Territories  ? 

48.  What  was  the  whole  population  of  the  U.  S.  in  1880  ? 

Note. — The  above  is  the  neiD  grouping  of  the  States  and  Territories 
proposed  by  the  Census  Bureau  of  1880. 

Q  U  ESTI  ONS. 

47.  What  is  Addition?  What  is  the  answer  called?  48.  Make  th.. 
sign  of  addition?  What  called?  How  read?  49.  Make  the  sign  of 
equality. 

53.  What  kind  of  numbers  can  be  added?  What  orders  ?  57.  How  do 
the  denominations  of  U.  S.  Money  increase  and  decrease?  58.  How  are 
thev  expressed?     60.  Give  the  general  rule.    Proof? 


»|UBTR  ACTION. 

Oral     Exercises. 

61.  1.   Edward  had  12  pears  in  his  basket  and  took  out  5 
of  them ;  how  many  were  left  ? 

Solution. — 5  pears  taken  from  13  pears  leave  7  pears.     Ans.  7  pears. 

2.  If  you  take  6  cents  from  14  cents,  how  many  will  be  left  ? 

3.  What  is  the  difference  between  7  pounds  and  15  pounds  ? 

4.  A  lady  bought  a  hat  for  $10  and  gave  in  payment  a  120 
bill ;  how  much  change  ought  she  to  receive  ? 

5.  James  is  16  years  old  and  his  sister  is  7 ;  what  is  the 
difference  in  their  ages  ? 

6.  If  9  yards  of  cloth  are  cut  from  a  piece  containing  24 
yards,  how  many  yards  will  be  left  ? 

7.  Charles  had  $25  silver  dollars  and  gave  8  of  them  to  the 
orphan  asylum  ;  how  many  dollars  did  he  then  have  ? 

8.  If  9  is  taken  from  17,  how  many  are  left  ? 

9.  How  many  more  is  18  than  6  ?     Than  7? 

10.  If  a  slate  cost  12  cents  and  a  reader  26  cents,  how  much 
more  will  one  cost  than  the  other  ? 

Definitions. 

62.  Subtraction  is  taking  one  number  from  another. 

63.  The  Subtrahend  is  the  number  to  be  subtracted. 

64.  The  Minuend  is  the  number  from  which  the  subtraction 
is  made. 

65.  The  Answer,  or  number  found  by  subtraction,  is  called 
the  Difference  or  Remainder. 

Note.— Subtraction  is  the  reveme  of  Addition.    Tlie  one  unites  numbers, 
the  other  separates  them. 


Subtraction,  27 

66.  The  Sign  of  Subtraction  is  — .  It  is  called  minus,  which 
means  less.  When  placed  between  two  numbers  it  shows  that 
the  number  after  it  is  to  be  subtracted  from  the  one  before  it. 

Thus,  the  expression  12  —  5  =  7  is  read,  "  12  minus  5  equals  7,"  or 
"  is  equal  to  7,"  or  "  12  less  5  equals  7." 

67.  The  Parenthesis  (  ),  and  the  Vinculum  ,  respec- 
tively show  that  the  numbers  included  by  them  are  to  be  con- 
sidered as  one  number. 

Thus,  16  —  (4  +  3)  shows  that  the  sum  of  4  and  3,  or  7  is  to  be 
subtracted  from  16,  and  the  result  is  9. 


How  many  are 

11.  14—6? 

17. 

16-5? 

23. 

23-7? 

29. 

52-6? 

12.  16  —  7? 

18. 

15  —  7? 

24. 

27  —  9? 

30. 

63  —  7? 

13.    11-6? 

19. 

18—4? 

25. 

34—6? 

31. 

74-8? 

14.   13—5? 

20. 

19-8? 

26. 

42-8? 

32. 

83-5? 

15.  15-8? 

21. 

18-9? 

27. 

35—6? 

33. 

97-8? 

16.  17—9? 

22. 

17  —  8? 

28. 

44—8? 

34. 

84—9? 

Development    of    JPrin c if l  e s . 

1.  What  is  the  difference  between  15  pencils  and  9  pencils  ? 

2.  What  is  the  difference  between  2  books  and  5  chairs  ? 
Ans.  Books  and  chairs  are  unlilce  units,  and  one  cannot  be 

subtracted  from  the  other. 

3.  Wliat  is  the  difference  between  9  units  and  15  units  ? 
Between  5  tens  and  3  tens  ? 

4.  What  is  the  difference  between  5  tens  and  3  ones  ? 

Ayis.   Tens  and  ones  are  unlike  orders  of  units,  and  one  cannot 
be  subtracted  from  the  other. 

5.  If  the  minuend  is  14  and  the  subtrahend  8,  what  is  the 
remainder  ? 

6.  If  the  remainder  6  is  added  to  the  subtrahend,  to  what  is 
the  sum  equal  ? 

Ans.  To  the  minuend. 

7.  What  is  the  difference  between  8  and  12  ?    If  you  add  3 
to  each  of  the  numbers  8  and.  12,  what  is  the  remainder  ? 

Ans.  4,  the  same  as  before. 


28  Snhtvdction. 

68.  From  the  examples  above  we  infer  the  following 

Principles. 

i°.  Only  like  numhers  and    like  orders  of  units    can  he 
subtracted  one  from  the  other, 

^°.  Th.e  difference  and  subtrahend  are  equal  to  the  minuend. 

S°.  If  two  numbers  are  equally  increased,  their  difference  is 
not  altered. 

Written     Exercises. 

69.  When  each  order  i7i  the   Subtrahend  is  less  than  the 
correspo7iding  order  of  the  Minuend. 

1.  From  4678  subtract  1435. 

Explanation, — We  write  the  subtrahend   under         operation. 
the  minuend,  placing  units  of  the  same  order  in  the  4678     Min. 

same  column.     Beginning  at  the   right,  we  say,  "5  1435      Sub 

units  from  8  units  leave  3  units,"  and  write  the  3  in 

units  place,  under  the  figure  subtracted.    Next,  3  tens  o/c4o     Rem. 

from  7  tens  leave  4  tens,  which  we  write  in  tens  place.  4  hundreds  from 
6  hundreds  leave  2  hundreds,  which  we  write  in  hundreds  place.  Finally 
1  thousand  from  4  thousand  leave  3  thousand,  which  we  write  in 
thousands  place.     The  remainder  is  3243. 

To  prove  the  result,  add  the  remainder  to  the  subtrahend,  and  if  the 
sum  is  equal  to  the  minuend  the  work  is  right.     (Art.  68,  2°.) 

Subtract  the  following  in  like  manner  : 

(2.)  (3.)  (4.)  (5.)  (6.) 

From        5374  6487  7636  8768  9689 

Take  2142  3243  4212  5243  6476 

7.  A  farmer  having  876  acres  of  land,. sold  375  acres;  how 
many  had  he  left  ? 

8.  A  man  having  a  note  of  $2365  due,  had  only  11231  on 
hand ;  how  much  more  must  he  collect  to  pay  the  note  ? 

9.  The  population  of  Cal.  in  1880  was  864,694,  that  of  Neb. 
was  452,402;  what  was  the  difference  ? 


Suhtraction,  29 


Oral     Exercises. 

1.  Charles  picked  10  quarts  of  chestnuts,  and  on  his  way 
home  sold  4  quarts.  The  next  day  he  picked  9  quarts  more  ; 
how  many  quarts  had  he  then  ? 

2.  John  had  15  cents  and  his  father  gave  him  10  more;  he 
then  spent  6  cents  for  candy  ;  how  many  cents  had  he  left? 

3.  A  man  having  $30,  spent  the  sum  of  $5  +  $4 ;  how  much 
had  he  left  ? 

How  many  are 

4.  i4_64-3?  10.  18  +  4—6?  16.  21  — (4  +  5)  ? 

5.  16  —  9  +  4?  11.  24  +  7—3?  17.  18  — (6  +  5)? 

6.  27  —  8  +  6?  12.  17  +  9  —  8?  18.  24— (7  +  8)  ? 

7.  23  —  7  +  4?  13.  28  +  4—6?  19.  32  — (5  +  7)?. 

8.  19  —  6  +  5?  14.  19  +  8  —  7?  20.  36  — (9+7)? 

9.  3i_8  +  6?  15.  25  +  6—9?  21.  38  — (8  +  10)? 

22.  Subtract  by  2's  from  40  to  0.     Thus,  40,  38,  36,  34,  etc. 

23.  By  2's  from  51  to  1.  27.  By  6's  from  51  to  0. 

24.  By  3's      "     60  to  0.  28.  By  7's      ''     64  to  0. 

25.  By  4's      ''     61  to  1.  29.  By  8's      ''     71  to  0. 

26.  By  5's      ^-      70  too.  30.  By  9's      ''     80  to  0. 

Written    Exercises. 

70.  ]V}ien  mi  order  ifi  tlie  SuUraliend  is  larger  than  the  cor- 
responding order  in  the  Minuend. 

1.  What  is  the  difference  between  5847  and  2563  ? 

Explanation. — Since  like  orders  only  can  be  sub-         operation. 
tracted,   for  convenience  we  write  them   under  each  5847     Min. 

other.     Beginning  at  the  right  we  say,  3  units  from  2563      Sub 

7  units  leave  4  units  ;  write  the  4  in  units  place.    Next,  — 

since  6  tens  are  more  than  4  tens,  we  take  one  of  the  O/vOi     "Rem. 

hundreds  (equal  to  10  tens),  and  add  it  to  4  tens,  making  14  tens ;  now 

6  tens  from  14  tens  leave  8  tens,  which  we  write  in  tens  place.     As  we 
took  1  from  8  hundreds,  only  7  hundreds  are  left,  and  5  hundreds  from 

7  hundreds  leave  2  hundreds,  which  we  write  in  hundreds  place.     Then, 
2  thousand  from  5  thousand  leave  3  thousand.  Ans.  3284. 


30  Subtraction. 

Notes. — 1.  The  process  of  taking  a  unit  from  a  higher  order  in  the 
minuend  and  adding  it  to  a  lower  order,  is  called  Borrowing  ten. 

2.  When  we  "borrow  10,"  it  is  more  logical  to  take  1  from  the  next 
order  of  the  miiauend  ;  but  practically  it  is  more  convenient  to  add  1  to  the 
next  order  of  the  subtrahend.     (See  Ex.  4.) 

Subtract  and  explain  the  following  in  like  manner : 

(2.)  (3.)  (4.)  (5.) 

From         22304  30426  60000  84357 

Take         12012  20343  32114  50018 

71.  Decimals,  and  dollars  and  cents  are  subtracted  like  inte- 
gers ;  the  decimal  point  in  the  remainder  being  placed  under 
those  in  the  given  numbers. 

,  6.  What  is  the  difference  between  1285.47  and  $159.30  ? 

Explanation. — Subtract   as  in   integers,   placing  $285.47 

the  decimal  point  in  the  remainder  under  those  in  the  159.30 

given  numbers. 

Ans.  $126.17 

(8.)  (9.)  (10.)  (11.) 

From        325.2  431.58  1562.67  16000.009 

Take        108^  249.39  320.48  2315.07 

72.  From  the  preceding  examples  we  derive  the  following 

GeneralRule. 

/.  Write  the  sivbtrahend  under  the  iiiiniiend  so  that 
units  of  the  same  order  stand  under  each  other. 

II.  Begin  at  the  right  and  subtract  each  order  sepa- 
rately, placing  the  remainder  heloiv. 

III.  If  any  order  of  the  subtraJiend  is  larger  than 
that  above  it,  add  ten  to  the  upper  order  and  subtract. 
Consider  the  next  order  of  the  minuend  one  less,  and 
proceed  as  before,  placing  the  decimal  point  in  the 
remainder  under  those  in  the  given  numbers.     (Art.  71.) 

Proof. — Add  the  remainder  to  the  subtrahend  ;  if  the 
sujy^    is    equal    to    the    minuejtd,    the    worh    is    right. 


Subtraction,  31 


AtPIjICATIONS, 

1.  A  man  bought  a  piece  of  cloth  containing  237  yds.,  and 
sold  124  yds.  of  it.     How  much  had  he  left  ? 

2.  A  merchant  had  on  hand  a  quantity  of  flour,  for  which 
he  asked  $245  ;  but  for  cash  he  sold  it  for  %%^l  less.  How 
much  did  he  receive  for  his  flour  ? 

3.  In  a  certain  academy  there  were  357  scholars,  168  of 
whom  were  young  ladies.  How  many  gentlemen  were 
there  ? 

4.  A  farmer  raised  4879  bushels  of  wheat,  and  sold  387G 
bushels.     How  much  had  he  left  ? 

5.  A  farmer  raised  1389  bu.  of  wheat  one  year,  and  1763 
the  next.  How  much  more  did  he  raise  the  second  year  than 
the  first  ? 

6.  A  man  bought  a  house  and  lot  for  $5687.  The  house 
was  worth  $3698;  how  much  was  the  lot  worth  ? 

7.  Suppose  a  man's  income  is  $3268  a  year,  and  his  expenses 
are  $2789.     How  much  can  he  save  in  a  year  ? 

8.  K  a  man  has  $3290  in  real  estate,  and  owes  $1631,  how 
much  is  he  w^orth  ? 

9.  A  father  gave  his  son  $8263,  and  his  daughter  $5240  ; 
how  much  more  did  he  give  his  son  than  liis  daughter  ? 

10.  A  man  bought  a  farm  for  $9467,  and  sold  it  for  $11230  ; 
how  much  did  he  make  by  his  bargain  ? 

11.  If  a  man's  income  is  $10000  a  year,  and  his  expenses 
$6253,  how  much  will  he  lay  up  ? 

12.  4165  —  2340.  19.     45723  —  31203. 
.      13.     5600  —  3000.                    20.     81647  —  57025. 

14.  7246  —  4161.  21.  265328  —  140300. 

15.  8670  —  7364.  22.  170643  —  106340. 

16.  17265  —  13167.  23.  465746  —  241680. 

17.  21480  —  20372.  24.  694270  —  590395. 

18.  30671  —  26140.  25.  920486  —  500000. 

26.  The  captain  of  a  ship  having  a  cargo  of  goods  worth 
$29230,  threw  overboard  in  a  storm  $13216  worth ;  what  was 
the  value  of  the  goods  left  ? 


32  Suhtraction. 

27.  A  merchant  bought  a  quantity  of  goods  for  $12645,  and 
afterwards  sold  them  for  $13960  ;  how  much  did  he  gain  by 
his  bargain  ? 

.  28.   A  man  paid  $23645  for  a  ship  and  afterwards  sold  it  for 
$18260  ;  how^  much  did  he  lose  by  his  bargain  ? 

Perform  the  following  operations  in  tlie  order  indicated  : 

29.  275  +  317  —  87  +  49  4-95—216  +  342  —  07. 

30.  436  —  122  +  63  +  786  —  678  +  406—309  +  360. 

31.  639  —  250  +  873  +  67  —  19  +  476—506  +  1000. 

32.  4678  —  2500  +  6200  —  4004  +  502—625  +  1600—268. 

33.  6450  +  476—4578  +  5065  +  250  —  1000  +  608. 

34.  87200—463  +  225  + 1800  —  6200—75  +  98  +  2256. 

35.  7300  +  163-4005-85  +  2640-1375-23  +  867. 

36.  9640  +  9200  —  7000  —  75  +  4560  +  125—2000  +  485. 

37.  1452  +  325  +  684—  (631  +  845)  =  ? 

38.  4850  +  6300  — (800  +  3285)  =  ? 

39.  $256.62  +  ($64.50  — $20)  =  ? 

40.  $5278 +  $340.50  — ($480.40 +  $65.75)  =  ? 

Oral    Problems    for     Review. 

73.  1.  The  sum  of  two  numbers  is  26  and  one  of  them  is  7 ; 
wliat  is  the  other  ? 

2.  The  greater  of  two  numbers  is  24  anfl  the  difference  is  9  ; 
what  is  the  less  ? 

3.  When  the  greater  of  two  numbers  and  their  difference 
are  given,  how  find  the  less  ? 

4.  The  less  of  two  numbers  is  28  and  the  difference  is  12 ; 
what  is  the  greater  ? 

5.  When  the  less  of  two  numbers  and  their  difference  are 
given,  how  find  the  greater  ? 

6.  A  boy  having  75  cents,  spent  32  cents  for  toys;  how 
many  cents  did  he  then  have  ? 

Analyhts.- 75  iH  oqual  to  7  tcnH  and  5  iinitH  ;  and  ?>2  is  oqual  to  3  tens 
and  2  units ;  now  3  tens  from  7  tons  leave  4  tens  or  40,  and  2  units  from 
5  units,  leave  3  units,  wliieh  added  to  40  makes  43.  Ans.  45]  cts. 

Note. — When  the  numbers  in  subtraction  are  large,  it  is  advisable,  iu 
mental  operations,  to  be^n  at  the  hi^diest  orders,  as  in  addition. 


Subtract  ion.  33 

7.  If  the  price  of  a  history  is  90  cts.,  and  that  of  a  reader  is 
70  cts.,  what  is  the  difference  in  their  price  ? 

8.  A  fiu'mer  raised  SO  bu.  com  and  sold  50  bu. :  how  many 
bushels  had  he  left  ? 

9.  The  united  ages  of  two  persons  is  ^  years,  and  the 
younger  is  '^'^  :  what  is  the  age  of  the  older  ? 

10.  William  and  Chiirles  together  caught  5S  fish,  and 
William  caught  "27 ;  which  caught  the  more,  and  how  many? 

11.  In  a  school  of  So  pupils.  -IS  ai-e  girls  :  how  many  Ixiys  are 
there,  which  department  is  the  larger,  and  by  how  many  ? 

12.  A  lady  haviug  "2  ten-dollar  bills,  paid  19  for  a  hat.  -$4  for 
lace,  and  $'2  for  gloves  :  how  much  money  had  she  left  ? 

13.  Which  is  the  greater.  '24  -f  10.  or  5"2  —  9  ? 

14.  A  gentleman  paid  $1*2  for  pants,  19  for  a  Test,  and  #7 
for  boots  ;  he  paid  for  them  with  "2  ten-dollar  bills  and  2  fives  : 
how  much  change  should  he  have  ? 

15.  A  merchant  paid  #7S  for  a  case  of  goods,  and  ^5  freight : 
for  how  much  must  he  sell  them  to  make  #10  ? 

16.  If  you  have  $1:7  and  pay  |17  for  a  bicycle  and  $"2  for  a 
cap,  how  much  mouey  will  you  have  left  ? 

17.  A  lad  had  51  uuirbles  :  he  gave  away  •2S  and  found  5  : 
how  many  marbles  had  he  then  ? 

Oral  Drill  in  Adding  and  Subtracting. 

1.  To  5  add  6,  subtract  3.  add  7.  subtract  S,  add  4,  subtract 
7,  add  S,  subtract  3  :  what  is  the  result? 

XOTE. — ^YMle  the  teacher  dictares  the  example.  "*  To  5  add  6.  subtract 
3,"  etc..  the  pupils  thi  .k  11.  S.  15.  etc.  The  roiswer  mav  be  given  in  con- 
cert, or  bv  some  individual  designated  by  the  teacher. 

2.  From  15  take  6,  add  7,  take  S,  add  5,  take  6,  add  1. 
take  9,  add  10  :  result  ? 

3.  To  11  add  5,  take  7,  add  1,  take  3,  add  S,  take  5,  add  6, 
take  4,  add  9,  take  6  :  result  ? 

4.  How  many  are  *23  — 7  — 3— 1-flO— S-1-5  — 7-f  6  ? 

5.  How  many  are  7  +  9  — 10-f  0—1-^7  — S-4-9? 

6.  How  many  are  1'2  -^  0  — S  -h  1—3  —  '20—10  -f  5  ? 

7.  How  many  are  •2?— S  — 9— 10 -[-7—0— S-r 9  — 7  —  5  ? 


34  Subtraction, 

8.  How  many  are  23  —  6  +  11  —  8  +  9  —  6-1-4—7  +  8  ? 

9.  How  many  are  32  —  5  +  3  —  7  +  6  —  8  +  9  —  7  +  4—6? 

10.  How  many  are  35  +  8  —  7  +  6  —  4  +  8  —  7  +  5  —  8  +  12? 

11.  How  many  are  38+7  —  4  +  5  —  8  +  6  +  2  —  9  +  6? 

12.  How  many  are  28  +  4—7  +  6  —  9  +  8  —  9  +  10  ? 

Written    Problems    for    Review. 

74.  1.  The  minnend  is  3642.05,  and  the  difference  is  3202.8  \ 
what  is  the  subtrahend  ? 

2.  Two  brothers  commenced  business  at  the  same  time ; 
one  gained  $3678  in  five  years,  the  other  gained  12387.  How 
much  more  did  one  gain  than  the  other  ? 

3.  The  subtrahend  is  48206.5  and  the  difference  is  35206.2  ; 
what  is  the  minuend  ? 

4.  A  ship  having  a  cargo  valued  at  $100000,  was  overtaken 
by  a  storm,  and  $27680  worth  of  goods  were  thrown  overboard. 
How  much  of  the  cargo  was  saved  ? 

5.  A  gentleman  having  $1768  on  deposit,  gave  a  check  for 
$175  to  one  man,  to  another  for  $238.25,  and  to  another  for 
$369.50.     How  much  remained  on  deposit  ? 

6.  An  orchard  contained  120  apple  trees,«47  peach  trees,  and 
28  pear  trees.  Of  the  apple  trees  26  were  cut  down,  18  of  the 
peach  trees  died,  and  5  of  the  pear  trees  were  blown  down. 
How  many  trees  wTre  left  ? 

7.  A  gentleman  had  $2700  to  distribute   among  his  three 
sons.     To  the  eldest  he  gave  $825,  to  the  second  $785,  and  the" 
remainder  to  the  youngest.     How  much  did  the  youngest  son 
receive  ? 

8.  A  merchant  had  in  his  storehouse  6384  bushels  of  wheat, 
3752  bushels  of  corn,  4564  bushels  of  oats,  and  1384  bushels  of 
rye ;  it  was  broken  open  and  3564  bushels  of  grain  taken  out. 
How  many  bushels  remained  ? 

9.  If  a  man's  income  is  $4586  a  year,  and  he  spends  $384.86 
for  clothing,  $568  for  house  rent,  $784.75  for  provisions, 
$568.50  for  servants,  and  $369  for  traveling,  how  much  will 
he  have  left  at  the  end  of  the  year? 


Subtraction.  35 

10.  A  gentleman  left  a  fortune  of  $18864  to  his  two  sons 
and  one  daughter;  to  one  son  he  gave  $6389,  to  the  other 
$6984.     How  much  did  the  daughter  receive  ? 

11.  A  man  having  $7689,  invested  $689  in  railroad  stock, 
$500  in  a  woolen  factory,  and  $1250  in  bank  stock.  How 
much  had  he  left  ? 

12.  What  number  added  to  3645  makes  630712  ? 

13.  A  man  worth  $30000,  lost  a  store  by  fire  worth  $5000,  and 
goods  to  the  amount  of  $3578.     How  much  had  he  left  ? 

14.  From  twenty-five  thousand,  twenty-five,  take  28  hundred. 

15.  From  16  millions,  16  thousand,  take  16  hundred. 

16.  The  difference  between  185  billions,  and  185  millions  ? 

17.  What  number  mast  be  added  to  836.25  to  make  2323  ? 

18.  How  many  times  can  563  be  subtracted  from  2815  before 
the  latter  will  be  exhausted  ? 

19.  What  number  is  that,  from  which  if  you  take  42371,  the 
remainder  will  be  19289  less  176.05  ? 

20.  What  number  is  that,  from  which  if  you  take  18268,  the 
remainder  will  be  26017  —  17312? 

21.  What  number  is  that,  from  which  if  27239  be  taken,  the 
remainder  will  be  9897—3076.5  ? 

22.  A  says  to  B,  "  I  have  2675  sheep  " ;  B  replies,  ''  I  have  763 
less  than  you  "  ;  C  adds,  "  I  have  as  many  as  both  lacking  105." 
How  many  sheep  had  B  and  C  ? 

23.  The  sum  of  3  numbers  is  23257  ;  the  first  is  9277,  the 
second  is  1283  less  than  the  first ;  what  is  the  third  number  ? 

24.  The  population  of  the  U.  S.  in  1840  was  17069453,  in 
1880  it  was  50155783  ;  what  was  the  increase  in  40  years  ? 

Questions. 

62.  What  is  Subtraction?  63.  The  Subtrahend?  64  Minuend?  65. 
The  Answer?  66.  The  Sign  of  Subtraction?  What  called  ?  How  read? 
67.  For  what  are  the  Parenthesis  and  Vinculum  used?  68.  What  num- 
bers only  can  be  subtracted  ?     What  orders  ? 

68.  If  the  difference  of  two  numbers  is  added  to  the  less,  to  what  is  the 
sum  equal?  If  two  numbers  are  equally  increased,  how  is  their  difference 
affected?  71.  How  subtract  decimals  and  dollars  and  cents?  72.  Give  the 
general  rule.     How  is  subtraction  proved? 


Ul/riPLICATION. 


-[Sy 


-K- 


Mental    Exercises. 

75.     1.  What  will  3  pencils  cost,  at  4  cents  apiece  ? 

Analysis. — At  4  cts.  apiece,  3  pencils  will  cost  the  sum  of  4  cts.  +  4  cts. 
+  4  cts.,  or  4  cts.  taken  3  times,  whicli  are  12  cts.  Or,  more  briefly, 
3  pencils  will  cost  3  times  as  much  as  1  pencil,  and  3  times  4  cts.  are 
13  cts. 

2.  At  15  each,  what  w^ill  4  hats  cost  ? 

3.  At  5  cts.  apiece,  what  will  3  bananas  come  to  ? 

4.  In  1  gallon  there  are  4  qts.;  how  many  qts.  are  in 
5  gallons  ? 

5.  At  6  cts.  a  lb.,  what  will  4  lbs.  of  rice  cost  ? 

6.  If  1  qt.  of  cherries  cost  6  cts.,  what  will  3  qts.  cost  ? 

7.  What  will  5  vests  cost,  at  $7  apiece  ? 

8.  If  it  takes  6  yds.  of  cloth  to  make  1  cloak,  how  many 
yards  will  it  take  to  make  5  cloaks  ? 

9.  In  1  week  there  are  T  days;  how  many  days  in  4  weeks? 
10.  How  many  units  in  five  8's  united  in  one  number  ? 


Definitions. 

76.  Multiplication  is  finding  the  amount  of   one  number 
taken  as  many  times  as  there  are  units  in  another. 

77.  The  Multiplicand  is  the  number  to  be  multiplied. 

78.  The  Multiplier  is  the  number  by  which  we  multiply. 
It  shows  how  many  times  the  multiplicand  is  to  be  taken. 

79.  The   Ansioer,  or  number   found   by   multiplication,  is 
called  the  Product. 

Thus,  when  it  is  said  that  4  times  6  are  24,  6  is  the  mutiplicand,  4  the 
multiplier,  and  24  the  product. 


Multiplication, 


37 


80.  The  multiplicand  and  multiplier  which  produce  the 
product,  are  called  its  Factors. 

81.  The  Sign  of  Multiplication  is  x .  It  shows  that  the 
numbers  between  which  it  is  placed  are  to  be  multiplied 
together,  and  is  read  ''  times/'  or  "  multiplied  by." 

Thus,  7  X  4  =  28,  is  read,  "  7  times  4,"  or  "  7  multiplied  by  4  equals  28." 

Note. — Multiplication  is  similar  in  principle  to  addition,  and  may  be 
performed  by  it.  Thus,  the  product  of  three  times  4,  is  12,  which  is  the 
same  as  the  sum  of  4  +  4  +  4, 

Multiplication    Table. 


2  times 

1  are   2 

2  '^     4 

3  "     6 

4 

5 
6 

7 


9 
10 

1 1 

12 


a 
a 

a 
a 


10 
12 

14 
16 
18 
20 
22 
24 


3  times 

1  are  3 

2  '' 

3  " 


4 

5 
6 

7 
8 

9 
10 

II 

12 


6 

9 

12 

15 
18 

21 

24 

27 

30 


4 

times 

I 

are  4 

2 

"     8 

3 

"   12 

4 

S 

^^   16 

''  20 

6 

7 

-  24 
''  28 

8 

"  32 

9 

0 

"  36 

^^  40 

I 

2 

"  44 
-  48 

5  times 

1  are  5 

2  '^  10 

20 

25 
30 

35 

8  ''  40 

9  "  45 

10  ''  50 

11  "  55 

12  '^  60 


6ti] 

Qies 

7ti 

I  are  6 

I  ai 

2  '' 

12 

2  ' 

3   " 

18 

3   ' 

4  " 

24 

4  ' 

5  ' 

6  ^ 

30 
36 

5  ' 

6  ^ 

r: 

42 
-  48 

7  ' 

8  ' 

9  ' 
10  ^ 

54 
'  60 

9   ' 

10  ' 

II  ' 

66 

II    ' 

12  ^ 

72 

12   ' 

14 

21 

28 

35 
42 

49 
56 

63 
70 

77 
84 


8  times 
I    are 
2 

3 

4 

5 
6 

7 
8 

9 
10 

II 

12 


16 

24 

32 
40 

48 
56 
64 
72 
80 
88 
96 


9  times 


I 

2 

3 

4 

5 
6 

7 
8 

9 

10 
1 1 
12 


are 


a 
a 
a 
a 
a 
a 
a 
a 


9 
18 

27 
3^ 
45 
54 
63 
72 
81 
90 

99 

108 


10  times 
I 

2 

3 
4 

5 
6 

7 


9 

10 
II 
12 


are    10 

''     20 

"     30 

"     40 

"     50 

60 

70 

80 

90 

100 


2 

3 

4 

5 
6 

7 
8 

9 

TO 


I 10   I    II 

120       12 


times 

are 

II 

iC 

22 

ii 
ii 

33 
44 

ii 

55 
66 

ii 

ii 

77 
88 

i( 

ii 

99 
no 

ii 

121 

a 

132 

12  times 


I 

2 

3 

4 

5 
6 

7 
8 

9 
10 

II 

12 


are 


ii 
ii 
ii 
ii 
a 
ii 
ii 


I  2 
24 

36 

48 

60 

72 

84 

96 

108 

120 

132 

144 


Note. — Promiscuous  exercises  upon  the  table  should  be  repeated  till 
any  combinations  within  its  limits  can  be  answered  instantly. 


38  Multvplication, 

11.  Count  by  3's  to  30  and  back  to  0. 

12.  Name  the  products  by  4  to  40  and  back. 

13.  Name  the  products  by  5  to  50  and  back. 

14.  Name  the  products  by  6  to  60  and  back. 

15.  Name  the  products  by  7  to  70  and  back. 

16.  Name  the  products  by  8  to  80  and  back. 

17.  Name  the  products  by  9  to  90  and  back. 

18.  How  many  times  7  are  28  ?  22.  Times  5  are  30  ? 

19.  How  many  times  6  are  42  ?  23.  Times  7  are  56  ? 

20.  How  many  times  8  are  48  ?  24.  Times  9  are  54  ? 

21.  How  many  times  7  are  63  ?  25.  Times  9  are  72  ? 

26.  6x7—5x6=?     29.   7x8—4x6?      32.  8x9  —  7x5? 

27.  7x5—4x8=?     30.   8x6  —  6x8?      33.   7x8  —  6x9? 

28.  5x9  —  6x7=?     31.   9x7  —  6x9?      34.   9x9-8x8? 

DeveTjOpment  of  Principtes. 

82.  1.  What  is  the  product  of  17  multiplied  by  4  ? 

2.  What  is  the  product  of  9  multiplied  by  5  ? 

3.  What  kind  of  numbers  are  these  miiUipUcands  9 
Ans.  The  first  is  concrete,  the  second  is  abstracts 

4.  What  kind  of  a  number  is  the  ynultiplier  f 
Ans.  It  is  an  abstract  number  in  both  examples. 

5.  What  kind  of  a  number  is  the  product  ? 
A71S.  The  same  in  each  as  the  multiplicand. 

6.  What  is  the  product  of  7  days  multiplied  by  9  pounds  ? 
Ans.  Pounds  are  concrete   numbers   and   7  days  cannot  be 

taken  4  pounds  times. 

7.  Which  is  the  greater  number,  3  times  4,  or  4  times  3  ? 

83.  From  the  above  examples  we  deduce  these 

Principles. 

i°.   The  multiiMcand  may  be  either  abstract,  or  coyicrete. 
2°.    The  multiplier  ynust  be  considered  an  abstract  number. 
S°.    The  midtiplicand  and  product  are  like  numbers. 
Jf°.   The  product  is  the  same  in  ivhatever  order  the  factors  are 
taken. 


Multiplicatioiu  39 

Written    Exercises. 

84.  When  the  multiplier  has  but  one  figure. 

1.  If  a  rail-car  goes  538  miles  a  day,  how  far  will  it  go  in 
4  days  ? 

Analysis. — The  car  ^vill  go  4  times  as  far  in  operation. 

4  days  as  in  1  day.     Write  the  multiplier  under  538      Multiplicand, 

the  multiplicand,  and  beginning  at  units  say,  ^      Multiolier 

"4  times  8  units  are  32  units,  or  8  tens  and  

3  units."  We  set  the  2  in  units  place  and  ^^^^'  ^^^^  miles, 
add  the  8  tens  to  the  product  of  tens,  as  in  addition.  Next,  4  times  3  tens 
are  12  tens,  and  8  tens  added  make  15  tens,  or  1  hundred  and  5  tens.  We 
write  the  5  in  tens  place,  and  add  the  1  hundred  to  the  product  of 
hundreds.  Finally,  4  times  5  hundreds  are  20  hundreds,  and  1  hundred 
added  makes  21  lumdreds,  or  2  thousand  and  1  hundred.  We  write  the  1 
in  hundreds  place,  and  the  2  in  thousands  place.  The  product  is  2152 
miles,  Aiu. 

2.  If  1  bale  of  cotton  weighs  250  pounds,  what  will  7  bales 
weigh  ? 

3.  A  drover  bought  6  flocks  of  sheep,  the  average  number 
of  which  was  735  ;  how  many  sheep  did  he  buy  in  all  ? 

(8.) 

5178  in. 
4 

85.  When  the  multiplicand  or  multiplier  contains  Decimals. 

Decimals  and  dollars  and  cents  are  multiplied  like  integers,  as  many 
figures  being  pointed  off  for  decimals  in  the  product,  as  are  found  in  both 
factors. 

9.  What  is  the  product  of  $64,375  multiphed  by  7  ? 

Explanation. — U.  S.  Money  as  well  as   Decimals  is  operation. 

multiplied  like  whole  numbers  ;   from   the  rio-ht   of  the  $64,375 

product,  as  many  decimal  figures  are  pointed  off  as  there  f^ 

are  decimal  places  in  hoth  factors.  

$450,625 

(10.)  (11.)  (12.)  (13.)  (14.) 

384.9  67.02  54.37  8.603  87.46 

5  6  8  7  9 


(4.) 

(5.) 

(6.) 

(7.) 

574  lbs. 

725  ft. 

869  yds. 

4256  bu, 

3 

4 

5 

3 

40  Midtiplication. 


(15.)                     (16.) 
$4352.67          $676,238 

(17.) 

$7283.04 

(18.) 

$9280.23 

(19.) 

$807,206 

6                        5 

7 

8 

9 

(20.) 
Multiply    $34.56 
By                       4 

(21.) 
$242.63 
6 

(22.) 

$0.96 
8 

(23.) 

$0,873 
9 

24.  What  cost  8  barrels  of  flour,  at  $7.50  a  barrel  ? 

25.  What  will  458  hats  cost,  at  $6  apiece  ? 

Note.— In  tliis  example  the  true  multiplicand  is  $6.     But  it  is  more 

convenient  to  use  6  as  tbe  multiplier  and  458  as  the  multiplicand,  as 

follows:     (Art.  83,  4  .) 

458 
Analysis. — At  $1  each,  458  hats  would  cost  $458,  and 

at  $6  each,  they  will  cost  6  times  $458,  or  $2748.  . - 

Ans.  $2748 

26.  What  cost  375  tons  of  hay,  at  $8  per  ton  ? 

27.  What  cost  5265  bales  of  cotton,  at  $8  a  bale  ? 

28.  At  $4  a  barrel,  what  cost  1500  barrels  of  apples  ? 

29.  At  $5.67  a  yard,  what  cost  8  yds.  of  cloth? 

30.  What  cost  2350  clocks,  at  $9  each  ? 

Mental    Exercises. 

86.     1.   What  will  4  vests  cost,  at  $7  apiece  ? 

1st.  Analysis. — 4  vests  will  cost  4  times  as  much  as  1  vest,  anc* 
4  times  $7  are  $28,  Ans. 

2d.  Analysis.— Since  1  vest  costs  $7,  4  vests  will  cost  4  times  $7, or  $28. 
Therefore,  4  vests  will  cost  $28. 

Note, — The  essentials  of  a  good  iVnalysis  are  clearness,  brevity,  appro- 
printe  language,  and  a  logical  conclusion.  Sameness  in  form  should  be 
avoided. 

2.  If  you  write  8  lines  a  day,  how  many  lines  will  you  write 
in  6  days  ? 

3.  If  there  are  5  school  days  in  1  week,  how  many  school 
days  are  there  in  12  weeks  ? 

4.  What  is  the  cost  of  9  bananas,  at  6  cents  each  ? 

5.  A  grocer  sold  a  turkey  weighing  8  lbs.  at  11  cts.  apound  ; 
what  did  it  come  to  ? 


Multiplication.  41 

6.  What  will  9  qts.  of  cherries  come  to,  at  12  cts.  a  quart  ? 

7.  At  8  cts.  a  mile,  what  will  it  cost  to  ride  9  miles  ? 

8.  Carrie  made  9  bouquets,  each   having  10  flowei'S  ;  how 
many  flowers  had  they  all  ? 

9.  At  $12  a  hundred,  what  will  4  hundred  melons  cost? 

10.  What  cost  8  dozen  eggs,  at  12  cts.  a  dozen  ? 

11.  If  a  man  walk  47  miles  a  day,  how  many  miles  will  he 
walk  in  6  days  ? 

Analysis. — 47  equals  4  tens  and  7  units.  Now  6  times  4  tens  are 
24  tens,  or  240,  and  6  times  7  units  are  42  units,  which  added  to  240  make 
282.     Ans.  282  miles  in  6  days. 

Note, — When  the  multiplicand  is  large,  it  is  advisable  in  mental  opera- 
tions to  begin  with  the  liigest  order,  as  in  addition.     (Art.  54.) 

12.  What  cost  4  acres  of  land,  at  $36  an  acre? 

13.  At  $75  a  share,  what  will  7  shares  of  Bank  Stock  come  to  ? 

14.  A  furniture  dealer  sold  10  sofas  at  $87  apiece  ;  what  did 
he  get  for  all  of  them  ? 

15.  If  a  railroad  car  goes  at  the  rate  of  57  miles  an  hour,  how 
far  will  it  go  in  8  hours  ? 

16.  How  many  are  9  times  73  ?     8  times  63  ? 

17.  How  many  are  7  times  86  ?     6  times  97? 

18.  How  many  are  8  times  94  ?     9  times  89  ? 

19.  If  melodeons  are  $75  apiece,  what  will  6  cost  ? 

Written    Exercises. 
87.     When  the  multiplier  has  two  or  more  figures. 
1.   What  is  the  product  of  324  multiplied  by  132  ? 

Explanation. — We   write   the  multiplier  under         324  Multiplicand, 
the  multiplicand,  as  in  the  margin,  and  beginning  at         ^^^2  Multiclier 
tlie  right,  multiply  by  each  order  successively.  

Thus,  multiplying  824  by  2  units,  the  first  partial         ^^^  P^^-  P^o<i- 
product  is  648,  the  right  hand  figure  of  which  we  set  in       972       "       " 
units  place.     Next,  multiplying  by  3  tens,  the  second    324  "       " 

partial  product  is  972  tens,  the  right  hand  figure  of 
which  we  write  in  tens  place,  under  the  multiplying    ^'^^^^  Entire  Prod, 
figure.     Finally,  the  tliird  partial  product  is  324  hundreds,  the  right  hand 
figure  of  which  we  place  under  the  order  which  produced  it. 

Adding  these  parti(d  products,  tlic  sum  42768,  is  the  entire  prodfict. 


42  Midti'plication. 

Note. — To  prove  the  correctness  of  the  work,  we  multiply  the  multiplier 
132  by  324  the  multiplicand,  and  as  the  two  results  agree,  the  work  is 
correct. 

Multiply  and  explain  the  following  in  like  manner: 

(2.)             (3.)                (4.)  (5.)  (6.) 

42.5          $563          $6.42         42.6  lbs.  678  yds, 

34         __42         27         23.4  346 

88.  From  the  preceding  exercises  we  derive  the  following 

General     Rule. 

/.  Write  the  multiplier  under  the  multiplicand,  and> 
beginning  at  the  right,  multiply  each  order  of  the  multi- 
plicand by  each  order  of  the  multiplier,  placing  the  right 
hand'  figure  of  each  partial  product  under  the  order 
which  produced  it. 

II.  Add  the  partial  products  together,  and  from  the 
right  point  off  as  many  figures  for  decimals  as  there  are 
places  of  decimals  in  the  multiplicand  and  multi- 
plier;  the  result  will  be  the  product  required. 

Proof. — Multiply  the  multiplier  by  the  multiplicand  ; 
if  the  two  results  a.gree  the  luorh  is  correct. 

(For  proof  by  casting  out  the  9's,  see  Art.  876,  Appendix. ) 

A.PJPT.ICATIONS. 

1.  Allowing  365  days  to  a  year,  how  many  days  has  a  man 
lived  who  is  45  years  old  ? 

2.  If  a  garrison  consume  725  pounds  of  beef  in  one  day, 
how  many  pounds  will  they  consume  in  126  days? 

3.  What  cost  243  chests  of  tea,  at  137  per  chest? 

4.  A  man  bought  268  horses,  at  $63  apiece  ;  what  did  they 
come  to  ? 

5.  If  sound  moves  1142  feet  in  a  second,  how  far  will  it  move 
in  60  seconds  ? 

6.  If  a  cattle  train  has  23  cars  and  each  car  contains  68 
sheep,  how  many  sheep  in  the  train? 


Multiplication.  43 

7.  How  much  can  a  man   earn  in  48  months  at  $125   a 
month  ? 

8.  At  $32  each,  how  much  will  it  cost  to  furnish  the  outfit 
for  560  policemen  ? 

9.  How  many  bushels  of  corn  may  be  raised  on  485  acres 
which  average  37  bu.  to  the  acre  ? 

10.  There  are  640  acres  in  a  square  mile ;  how  many  acres 
are  there  in  75  square  miles  ? 

What  is  the  product  of 

11.  8623  by  24?  23.  2734  by  234? 

12.  2538  by  39  ?  24.  4803  by  .325  ? 

13.  4752  by  43  ?  25.   6578  by  467  ? 

14.  5843  by  63  ?  26.  5967  by  504  ? 

15.  $32.45  by  57?  27.  43672  by  564  ? 

16.  $47.08  by  68?  28.  54865  by  647  ? 

17.  6264  by  70?  29.   60435  by  704? 

18.  $29,451  by  49  ?  30.   74321  by  839  ? 

19.  420643  by  76  ?  31.   543267  by  1563  ? 

20.  572062  by  84?  32.   684039  by  1783? 

21.  398025  by  87  ?  33.   709564  by  2803  ? 

22.  703270  by  93  ?  34.   894037  by  3085  ? 

35.  Tf  a  clerk  has  $36  a  month  for  the  first  4  months  ;  $48 
a  month  for  the  next  4  ;  and  $60  a  month  for  the  next  4;  what 
will  he  receive  for  the  year  ? 

36.  If  I  receive  $350  a  month,  how  much  shall  I  have  at  the 
end  of  the  year,  after  deducting  $38  a  month  for  board  ? 

37.  If  it  takes  385  laborers  18  months  to  build  a  railroad, 
how  long  would  it  take  1  man  to  build  it  ?  ^ 

38.  A  ship  of  war  has  provisions  to  last  a  crew  of  645  men 
90  days  ;  how  long  would  they  last  1  man  ? 

39.  If  I  sell  29  bbls.  of  flour  at  $8  a  barrel,  and  50  bbls.  of 
beef  at  $18  a  barrel,  and  receive  5  hundred-dollar  bills  in  pay- 
ment, how  much  will  be  due  me  for  both  ? 

40.  A  farmer  sold  32  sheep  at  $8  a  head,  and  9  cows  for  $45 
apiece  ;  how  much  more  did  he  receive  for  the  cows  than  the 
*heep  ? 

41.  What  cost  169  chairs,  at  $3.25  apiece  ? 


44  Midtiplication. 

42.  What  cost  279  barrels  of  salt,  at  11.75  a  barrel? 

43.  What  cost  1565  acres  of  land,  at  127  per  acre  ? 

44.  What  cost  758  baskets  of  peaches,  at  12.50  a  basket? 

45.  If  a  hall  Avill  seat  1250  persons,  and  each  seat  is  occupied 
by  a  person  weighing  135  lbs.  what  weight  is  sustained  by  the 
floor  ? 

46.  Bought  2  farms;  one  contained  327  acres  at  $83  an 
acre,  the  other  526  acres  at  $58  an  acre.  What  did  they  both 
cost  ?     AVhat  was  the  difference  in  their  cost  ? 

47.  What  is  the  value  of  56  railway  cars,  at  $9550.75  each  ? 

48.  A  man  bought  31  colts  at  $28  apiece,  and  paid  10  tons 
of  hay  at  125  a  ton  ;  how  much  did  he  ow^e  for  the  colts? 

49.  A  grocer  bought  21  barrels  of  flour  at  |5  per  barrel,  and 
sold  16  barrels  of  it  at  17;  finding  the  rest  damaged,  he 
put  it  at  13  a  barrel.  How  much  did  he  make  or  lose,  by  the 
operation  ? 

50.  A  farmer  having  75  turkeys,  sold  50  of  them  at  86  cents 
apiece,  and  the  rest  at  54  cts.  apiece  ;  what  did  they  come  to  ? 

51.  A  man  owns  7  orchards  ;  in  each  orchard  there  are  8  rows 
of  apple-trees,  and  29  trees  in  each  row;  how  many  apple-trees 
has  he? 

52.  In  a  certain  school  there  are  3  departments,  in  each 
department  there  are  11  classes,  and  in  each  class  there  are  48 
pupils ;  how  many  pupils  are  there  in  the  school  ? 

53.  If  you  multiply  58  by  37,  and  this  product  by  29,  what 
will  be  the  result  ? 

54.  If  450  is  multiplied  by  254,  and  this  product  by  178, 
what  will  be  the  result  ? 

55.  A  man  sent  37  loads  of  wheat  to  market;  every  load 
contained  16  bags,  and  each  bag  3  bushels  ;  how  many  bushels 
did  he  send? 

56.  What  is  the  product  of  378  +  342  by  (763  —  251)  ? 

57.  What  is  the  product  of  254  +  451  by  (836  —  434)? 

58.  Required  the  product  of  823  —  567  by  (827  +  230). 

59.  Multiply  267  +  75  +  430  by  (468  —  324). 

60.  Multiply  869  —  675  by  (300  +  87  +  90). 

61.  What  is  the  product  of  (843  —  478)  x  (973  +  379)  ? 


Multiplication.  45 

89.  To  multiply  by  the  Factors  of  the  Multiplier. 

1.  What  will  15  tables  cost  at  $7  apiece  ? 

OPERATION. 

Analysis. — The  factors  of   15   are  5   and  3.  ^'^  Cost  of  i  Table, 

(Art.  80.)     Now   as  1    table   costs   $7,5   tables  k 

will    cost   5   times    $7   or    $35.      Again,    since 

15  =  3  X  5  it  follows  that   15  tables  will  cost  3  $35      "     "  5      " 

times  as  much  as  5  tables,  and  3  times  $35  are  3 

$105.     Hence,  the  777^ 

^lUo     '•     '•  15    '• 

Rule. — Jlicltiply  the  niulUplicaiid  by  one  of  the  factors 
of  the  inultiplier,  then  this  product  by  another,  and  so 
on,  till  all  the  factors  have  been  used. 

The  last  product  will  be  the  answer. 

2.  What  will  27  sofas  cost,  at  $85  apiece  ? 

3.  What  will  24  wagons  cost,  at  S3 7  apiece? 

4.  What  will  36  cows  cost,  at  $19  per  head  ? 

5.  If  a  man  travels  at  the  rate  of  42  miles  a  day,  how  far 
can  he  travel  in  205  days? 

6.  What  cost  45  acres  of  land,  at  110  dollars  -per  acre  ? 

7.  At  $6  per  week,  how  much  will  it  cost  a  person  to  board 
52  weeks  ? 

8.  At  the  rate  of  56  bushels  per  acre,  how  much  corn  can 
be  raised  on  460  acres  of  land  ? 

9.  What  cost  672  pieces  of  cashmere,  at  $24  apiece  ? 
10.  What  cost  1265  yoke  of  oxen,  at  $72  per  yoke  ? 

90.  When  the  Multiplier  has  ciphers  on  the  right. 

1.  What  is  the  product  of  56  multiplied  by  10  ? 

Solution. — When  a  figure  is  moved  one  'place  to  the  left,  its  value  is 
increased  ten  times.  (Art.  34,  3'.)  Hence,  if  we  annex  a  cipher  to  56  we 
mnltiply  it  by  10  and  it  becomes  560.  Ans.  560. 

2.  Multiply  64  by  100. 

Solution. — Annexing  two  ciphers  to  64  increases  its  value  100  times, 
and  therefore  multiplies  it  by  100.  Ans.  6400. 

3.  Multiply  87  by  1000.    A7is.  87000. 


46 


Multiplication. 


4.   Multiply  316  by  40. 

Explanation.— Multiplying  316  by  4  ones,  it  becomes 
1364.  But  we  are  required  to  multiply  by  4  tens  instead 
of  4  ones;  therefore  the  true  product  is  ten  times  1264. 
To  correct  this  result,  we  annex  a  cipher  to  it,  which  mul- 
tiplies it  by  10. 


OPERATION. 

316 
40 


Ans.  12640 


5.   Multiply  345  by  700. 

Solution.— Multiplying  by  7  ones  only,  gives  2415, 
which  must  be  multiplied  by  100  for  the  true  product. 
Tliis  is  done  by  annexing  two  ciphers.     Hence, 


91.   To  multiply  by  10,  100,  1000,  etc. 


OPEEATION. 

345 

700 

Ans.  241500 


Annex  as  many  ciphers  to  the  inultiplicand  as  there 
are  ciphers  in  the  multiplier. 

When  the  significant  figures  have  ciphers  on  the  right. 

Multiply  by  the  significant  figures,  and  to  the  result 
annex  as  many  ciphers  as  are  on  the  right  of  both  fac- 
tors.   (See  Art.  864,  Appendix.) 

6.  What  will  100  bales  of  cotton  weigh,  at  468  lbs.  to  abate  ? 

7.  How  many  pages  in  2300  books,  of  352  pages  each  ? 

21.  56300000  X  64  =  ? 

22.  62300000  X  890  =  ? 


8.  476  X  1000  =  ? 

9.  53486  X  10000  =  ? 

10.  12046708  X  100000  =  ? 

11.  26900785x1000000=? 

12.  89063457x10000000=? 

13.  9460305068x100000=? 

14.  1920  x  2000  =  ? 

15.  4376  X  2500  =  ? 

16.  50634  X  41000  =  ? 

17.  630125  X  620000  =  ? 

18.  12000  X  31  =  ? 

19.  370000  X  32  =  ? 

20.  8120000  X  46  =  ? 


23.  54000000  X  700  =  ? 

24.  43000000  X  600  =  ? 

25.  563800  X  7200  =  ? 

26.  1230000  X  12000  =  ? 

27.  310200  X  20000  =  ? 

28.  2065000  X  810000  =  ? 

29.  2109090  X  510000  =  ? 

30.  6084201  X  740000  =  ? 

31.  7283900  x  958300  =  ? 

32.  86007400  x  9700  =  ? 

33.  90690000  X  8600  =  ? 


Multiplication.  47 

Oral    Problems    for    Review. 

92.  1.  If  4  men  can  do  a  job  of  work  in  6  days,  how  long 

will  it  take  1  man  to  do  it  ? 

Analtsis. — It  will  take  1  man  4  times  as  many  days  as  it  takes 
4  men ;   and  4  times  6  days  are  24  days,  Ans. 

2.  In  1  peck  are  8  quarts ;  how  many  quarts  in  6  pecks  ? 

3.  How  many  quarts  are  7  pecks  and  3  quarts  ? 

4.  In  1  bushel  are  4  pecks  ;  how  many  pecks  in  15  bushels  ? 

5.  How  many  pecks  in  20  bushels  and  3  pecks  ? 

6.  If  9  men  can  build  a  wall  in  20  days,  how  long  will  it 
take  1  man  to  do  it  ? 

7.  In  1  pound  are  16  ounces  ;  how  many  ounces  in  4  pounds? 

8.  How  many  ounces  in  5  lbs.  7  ounces  ? 

9.  If  a  barrel  of  flour  will  last  8  persons  12  days,  how  long 
will  it  last  1  person  ? 

10.  AVhat  will  8  lbs.  of  maple  sugar  cost,  at  9  cts.  a  pound  ? 

11.  How  many  inches  in  11  feet? 

12.  How  many  feet  in  12  yds.  and  2  feet? 

13.  How  many  quarts  in  25  gallons  of  milk? 

14.  How  many  quarts  in  30  gallons  and  5  quarts  ? 

15.  If  two  men  start  from  the  same  place  and  travel  in 
opposite  directions,  one  at  the  rate  of  4  miles  per  hour,  the 
other,  3  miles,  how  far  apart  will  they  be  in  6  hours  ? 

Written    Problems    for    Review. 

93.  17.  George  has  27  cents,  and  Henry  has  3  times  as 
many  cents  as  George  lacking  5  ;  how  many  cents  has  Henry  ? 
How  many  have  both  ? 

18.  At  a  military  parade  there  were  5  regiments,  in  each 
regiment  8  companies,  in  each  company  9  platoons,  and  in 
each  platoon  10  soldiers  ;  how  many  soldiers  were  on  parade  ? 

19.  If  325  men  can  grade  a  street  in  28  days,  how  long  will 
it  take  1  man  to  do  it  ?  How  much  will  he  receive  for  it,  if  he 
has  $2  per  day  ? 

20.  A  and  B  are  20  miles  apart,  and  travel  in  opposite 
directions,  K  goes  4  miles  an  hour  and  B  5  miles ;  how  far 
apart  will  they  be  in  48  hours  ? 


48  Multiplication, 

21.  If  I  hire  a  carpenter  at  128  a  month,  and  his  appren- 
tice at  $14,  how  much  will  be  due  them  in  12  months  ? 

22.  A  man  bought  a  drove  of  1560  sheep,  at  $4  a  head; 
it  cost  him  $68  to  send  them  to  market,  and  they  brought  him 
$5  apiece ;  how  much  did  he  make  on  them  ? 

23.  A  drover  bought  360  head  of  cattle  and  96  horses  ;  he 
afterwards  sold  the  former  at  a  profit  of  $19  a  head,  and  the 
latter  at  a  loss  of  23  dollars  a  head  ;  did  he  gain  or  lose  by  the 
operation,  and  how  much  ? 

24.  A  grocer  bought  585  barrels  of  flour  at  $6  a  barrel,  and 
117  barrels  at  $7  ;  he  then  sold  the  whole  at  $6.50.  What  was 
the  result  of  his  speculation  ? 

25.  In  music,  two  minims  equal  a  semibreve  ;  two  crotchets 
a  minim  ;  two  quavers  a  crotchet ;  two  semi-quavers  a  quaver  ; 
and  two  demi-semiquavers  a  semi-quaver;  how  many  demi- 
semiquavers  are  equal  to  259  semi-breves  ? 

26.  Two  persons  start  from  the  same  place,  and  travel  in  the 
same  direction  ;  one  at  the  rate  of  33  miles  per  day,  and  the 
other  at  the  rate  of  37  miles  per  day ;  how  far  apart  will  they 
be  at  the  end  of  a  year  ? 

27.  Multiply  two  thousand  seven,  by  one  thousand  four. 

28.  Multiply  four  thousand  forty,  by  two  thousand  one  hun- 
dred three. 

29.  Multiply  forty  thousand,  four  hundred  four,  by  ten 
thousand  ten. 

30.  Multiply  one  hundred  five  thousand  seven,  by  sixty 
thousand,  four  hundred  three. 

31.  Multiply  five  millions,  two  hundred  six,  by  seventy 
thousand,  two  hundred  five. 

Questions. 

76.  What  is  Multiplication  ?  77.  The  Multiplicand  ?  78.  The  Multi- 
plier ?     What  does  the  Multiplier  show  ?     79.  What  is  the  Answer  called  ? 

80.  The  numbers  which  produce  the  product  called  ?  81.  Make  the  Sign 
of  Multiplication.     What  does  it  show  ?     How  read  ? 

85.  How  proceed  when  the  multiplicand  or  multiplier  has  decimals  ? 
88.  Give  the  general  rule  ?  How  prove  multiplication?  89.  How  multi- 
ply by  the  factors  of  a  number  ? 

91.  How  multiply  by  10,  100,  1000,  etc.?  When  there  are  ciphers  on 
the  right,  how  proceed  ? 


i^ 


IVISION 


■I-  ^ 


Oral     Exercises, 

94.  1.  How  many  times  can  4  cents  be  taken  from  a  purse 
containing  12  cents? 

Solution.— 13  cents  —  4  cts.  =  8  cts. ;    8  cts.  —  4  cts.  r=  4  cts. ;   and 

4  cts.  —  4  cts.  =  0.     Ajis.,  3  times. 

2.  How  many  times  are  4  cents  contained  in  12  cents  ? 

3.  How  many  3's  in  15  ?     How  many  5's  ? 

4.  How  many  oranges  at  4  cts.  apiece,  can  I  buy  for  20  cts.? 

Analysis. — I  can  buy  as  many  oranges  as  4  cents  are  contained  times  in 
20  cents.     Ans.,  5  oranges. 

5.  How  many  times  3  in  18?     How  many  times  6  ? 

6.  In  1  gallon  are  4  quarts  ;  how  many  gallons  in  24  quarts  ? 

7.  In  1  week  are  7  days  ;  how  many  weeks  in  28  days  ? 

8.  \Yhat  is  one  of  the  5  equal  parts  of  30  ? 

Analysis.— Since  30  is  6  times  5,  one  of  the  5  equal  parts  of  30  is  6. 

9.  On  Christmas  day  a  father  divided  $25  equally  among  his 

5  children  ;  how  many  dollars  did  each  receive  ? 

10.  How  many  boxes,  each  holding  12  lbs.,  will  a  dairyman 
require  to  pack  36  lbs.  of  butter  ? 

11.  A  teacher  having  45  pupils,  formed  them  into  classes  of 
9  each  ;  how  many  classes  did  she  have  ? 

Definitions. 

95.  Division  is  finding  hoiv  7nany  times  one  number  is  con- 
tained in  another;  or  finding  one  of  the  equal  i)arts  of  a 
number. 

96.  The  Dividend  is  the  number  to  be  divided. 

97.  The  Divisor  is  the  number  by  which  we  divide. 

98.  The  Answer,  or  number  found  by  division,  is  called  the 
Quotient.  It  shows  how  many  times  the  divisor  is  contained 
in  the  dividend. 


50  Division, 

99.  The  Remainder  is  the  part  of  the  dividend  left  when 
the  divisor  is  not  contained  in  it  an  e.ract  number  of  times,  and 
is  always  less  than  the  divisor. 

100.  The  Sign  of  Division  is  -^.     It  is  read  *' divided  by." 
Thus,  8  -4-  4  is  read,  "  8  divided  by  4." 

101.  Division  is  also  denoted  by  writing  the  cUviso?'  under 
the  dividend,  with  a  line  between  them. 

Thus,  I  is  read,  "8  divided  by  4." 

Notes. — 1.  Division  is  the  reverse  of  multiplication  ;  the  former  sepa- 
rates numbers  into  equal  parts  ;  the  latter  unites  equal  parts  in  one  num- 
ber. The  dividend  corresponds  to  the  product,  and  the  divisor  and  quo- 
tient to  the  multiplier  and  multiplicand.     (Art.  79.) 

2.  Division  is  also  similar  in  principle  to  subtraction,  and  may  be 
performed  by  it.  Thus,  5  is  contained  in  15,  3  times,  and  5  can  be  sub- 
tracted from  15,  3  times. 


12. 

How  many 

times  3  in  36  ? 

13. 

4  in  28? 

16. 

.     5  in  45  ? 

19. 

8  in  56  ? 

14. 

4  in  32  ? 

17, 

.     6  in  54  ? 

20. 

9  in  54? 

15. 

6  in  48  ? 

18. 

.     7  in  63  ? 

21. 

9  in  72? 

22. 

18-f-3  =  ? 

30. 

32^4=? 

38. 

36- 

^3=:? 

23. 

27^3  =  ? 

31. 

28-^4=? 

39. 

48- 

=-4=? 

24. 

35-^5  =  ? 

32. 

42 -=-7=? 

40. 

54- 

=-9r:r? 

25. 

45-^5  =  ? 

33. 

56-=-8=? 

41. 

56- 

=-8=? 

26. 

30-^6=? 

34. 

35^7=? 

42. 

72- 

=-9=? 

27. 

42-^6=? 

35. 

48-^8=? 

43. 

88- 

l-llrr^? 

28; 

60-^5r=? 

36. 

64^8=? 

44. 

72- 

1-8=? 

2^. 

54-f-6==? 

37. 

63-^7=? 

45. 

96- 

1-12=? 

46. 

-v-=? 

49. 

_5_4  — 
6    — 

:?             52. 

-¥-= 

=  p 

55. 

W=? 

47. 

48.  —  '? 
6 

50. 

63  — 

7    — 

:?             53. 

^■- 

—  9 

56. 

1  20 9 

1^" 

48. 

4f=? 

51. 

¥= 

:?            54. 

M= 

-9 

57. 

1  .3  3.  —  9 
-T3   —  • 

102.  The  Name  of  the  equal  parts  into  which  a  number  or 
thing  is  divided  depends  upon  the  Jtumher  of  parts.      Thus,^ 


Division.  51 

One  of  ttuo  equal  parts  is  called  One-half,  written  J. 

One  of  three  equal  parts  is  called  One-third,  written  J. 

One  of  four  equal  parts  is  called  One-fourth,  written  J. 

Two  of  three  equal  parts  are  called  Two-thirds,         written  f . 
Three  of  four  equal  parts  are  called  Three-fourths,  written  f. 

103.  Write  the  following  in  figures  : 

1.  Two-ninths  ;  three-fifths  ;  four-fifths  ;  five-eighths. 

2.  One-sixth  ;  two-sixths  ;  four-sixths  ;  five-sixths. 

3.  Two-sevenths;  three-sevenths;  fiye-sevenths. 

4.  Three-tenths ;  five-ninths  ;  three-elevenths  ;  six-twelfths. 

104.  AVhen  a  unit  is  divided  into  equal  parts,  the  parts  are 
called  Fractions. 

5.  What  part  of  3  is  1  ?    Is  2  ? 

Analysis. — If  3  is  divided  into  3  equal  parts,  one  of  these  parts  is 
l-third  of  3 ;  2  of  the  parts  are  2-tliirds  of  3. 

6.  What  part  of  4  is  1  ?     What  part  of  3  ? 

7.  What  part  of  5  is  2  ?    Is  3  ?    Is  4  ? 

8.  How  find  a  half  of  a  number  ?    A  third,  a  fourth,  etc. 
Ans.,  By  dividing  it  respectively  by  2,  3,  4,  etc. 

9.  What  is  1  third  of  6  ?     Of  9  ?     Of  12  ?     Of  18  ? 

10.  What  is  1  fifth  of  10  ?     Of  15  ?     Of  30  ?     Of  45  ? 

11.  What  is  1  eighth  of  24  ?     Of  32  ?     Of  40  ?     Of  48  ? 

12.  If  $40  are  distributed  among  8  laborers,  what  part  of  the 
money  and  how  many  dollars  will  each  receiye  ? 

Analysis.— One  is  l-eighth   of  8.      Therefore,    1   man   will  receive 
1 -eighth  of  $40,  which  is  $5,  Ans. 

13.  How  many  tons  of  coal,  at   $7   a  ton,  can  be  bought 
for  $63  ? 

14.  A  farmer  sold  5  tons  of  hay,  at  $12  a  ton,  and  took  his 
pay  in  flour  at  $6  a  barrel;  how  many  barrels  did  be  receive? 

15.  An  express  traveled  108  miles  in  9  hours  ;  at  what  rate 
was  that  per  hour  ? 

16.  How  many  times  is  \  of  42  contained  in  54? 

17.  How  many  times  is  \  of  63  contained  in  84  ? 


53  Division, 


jyErELOPMENT     OF     I^RINCIPLES. 

105.  1.  How  many  times  4  cents  in  20  cents  ? 

2.  What  kind  of  a  number  is  the  quotient  ? 
Ans.  It  is  an  ahdract  number. 

3.  If  30  apples  are  divided  into  5  equal  parts,  how  many 
apples  will  there  be  in  one  part  ? 

4.  What  kind  of  number  is  the  quotient  ? 

Ans.  A  concrete  number,  the  same  as  the  dividend. 

5.  If   7  is  the  divisor  and  4  the   quotient,   what  is   the 
dividend  ? 

Ans.  Their  product,  7x4,  or  28. 

106.  From  the  above  examples  we  derive  the  following 

Principles. 

i°.    When   the   divisor  and  dividend  are  lihe  numbers,  the 
quotient  is  an  abstract  number. 

2°.   When  the   divisor  is  an  abstract  number,  the    quotient 
and  dividend  are  like  numbers. 

3°.   The  product  of  the  divisor  and  quotient  is  equal  to  the 
dividend. 

Written     Exercises. 

107.  1.  Divide  952  by  4. 

Explanation.— Write  the  divisor  on  the  left  Divisor.  Divid.  Quotient, 
of  the  dividend,  witli  a  curved  line  between  them.  4  )  952  (  238 

First. — Beginning  at  the  left  to  divide,  we  find  g 

4  is  contained  in  9  hundreds  2  (hundreds)  times,  — 

and  place  the  2  (in  hundreds  place)  at  the  right  of  -'■" 

the  dividend  for  the  first  quotient  figure.  12 

Second. — We  multiply  the  divisor  by  the  quo-  oo 

tient  2,  and   set  the  remainder  under  the  order 
divided.  ?? 

Third. — Subtract  the  product  from  the  part  of  the  dividend  used. 

Fourth. — To  the  remainder  1  (hundred)  we  annex  the  5  tens,  making 
15  tens,  for  a  second  partial  dividend.  Now  4  is  contained  in  15  tens, 
3  (tens)  times  ;  write  the  3  in  tens  place  in  the  quotient,  and  multiplying 
the  divisor  by  it,  subtract  the  ])roduct  from  the  second  partial  dividend. 


Division,  53 

To  the  remainder  3  tens,  we  annex  the  3  units,  making  32  units,  a 
third  partial  dividend.  4  is  contained  in  32  units,  8  times.  Write  the  8 
in  units  place  in  the  quotient.  Multiplyhig  the  divisor  by  it  and  subtract- 
ing the  product,  there  is  no  remainder.     Ans.  238. 

Proof— Quotient  238  x  4  (Divisor)  =  952  (Dividend).     (Art.  106,  3°) 

2.  Divide  6570  by  5,  and  explain  the  operation  in  like  man- 
•ner.     Ans,  1314. 

3.  Divide  7650  by  6.  6.  Divide  9219  by  7. 

4.  Divide  8211  by  7.  7.   Divide  68696  by  6. 

5.  Divide  9872  by  4.  8.  Divide  89240  by  8. 

9.  What  is  the  quotient  of  6272  divided  by  4  ? 

Explanation. — We  draw  a  line  under  operation. 

the  dividend,  and  begin  to  divide  at  the  Divisor.  4  )  6272  Dividend, 
left  as  before.     Dividing  6  (thousands)  by  4,  T^c 

the  quotient  is  1  (thousand),  which  we  write  ^^° 

below  the  line,  under  the  order  divided.  Subtracting,  the  remainder  is  2 
(thousands).  To  this  we  annex,  mentally,  the  2  hundreds,  making  22 
(hundreds)  for  the  next  partial  dividend.  Divide  as  before,  and  proceed 
in  this  way  till  all  the  orders  are  divided,  carrying  the  multiplications  and 
subtractions  in  the  mind,  simply  setting  down  the  quotient  figures.  The 
quotient  is  1568. 

Proof.— The  quotient  1568  x  4  (divisor)  =  6272  (dividend). 

Divide  and  prove  the  following  in  like  manner  : 

(10.)  (11.)  (12.)  (13.) 

3  )  1134  5  )  1230  6  )  2562  7  )  8638 

(14.)  (15.)  (16.)  (17.) 

6  )  3276  8  )  9872  7  )  2345  9  )  3141 

108.  The  method  of  dividing  in  which  the  results  of  the 
several  steps  are  set  down,  as  illustrated  by  Ex.  1,  is  called 
Long  Division. 

109.  The  method  in  which  the  quotient  only  is  set  down, 
the  results  of  the  several  steps  being  carried  in  the  mind,  as 
illustrated  by  Ex.  9,  is  called  Short  Division. 

When  the  divisor  does  not  exceed  12,  this  method  is  preferable. 


54  Division. 

18.  Divide  13875  by  4. 

Notes. — 1.  To  indicate  the  division  of  the  final  operation. 

remainder,  if  any,  it  must  be  written  over  the  divi-  4  )  13875 

sor  and  placed  at  the  riffht  of  the  quotient  as  part  .  iTT^oo 

^  .,  Ans.  34681- 

OI   it.  * 

2,  In  proving  the  work,  the  remainder,  if  any,  must  be  added  to  the 
product  of  the  divisor  and  quotient.     (Art.  99.) 

* 
(19.)  (20.)  (21.)  (22.) 

5  )  4567  6  )  3971  8  )  6567  9  )  41756 


Solve  tlie  following,  both  by  Long  and  Short  Division  : 

(23.)  (24.)  (25.)  (26.) 

5  )  76453  4  )  82354  6  )  52387  7  )  63874 

(27.)         (28.)         (29.)         (30.) 

7  )  842952    6  )  428463    8  )  768345_   9  )  783952 

(31.)  (32.)  (33.)  (34.) 

8  )_6592^  9  )  75327         10  )  562340^      12  )  18396 

110.  Decimals  are  divided  like  integers,  and  from  the  i^ight 
'of  the  quotient  as  many  figures  must  be  pointed  off  for  deci- 
mals, as  the  decimal  places  in  the  dividend  exceed  those  in  the 
'divisor. 


OPERATION. 

9  )  391.86 


35.  Divide  391.86  by  9. 

Explanation. — We  divide  as  in  whole  numbers 
and  point  off  two  figures  for  decimals  in  the 
quotient.  Ans.    43.54 

36.  Divide  198.752  by  8.     Ans.  112.344. 

37.  Divide  2563.48  by  4.  38.   Divide  645.328  by  8. 

39.  At  14  a  yard,  how  many  yards  of  cloth  can  be  bought 
for  19850  ? 

40.  If  '1^48.78  are  divided  into   9  equal  parts,   what  is  the 
Talue  of  each  part  ? 


Division,  55 


Oral     Exercises. 

111.  1.  If  the  price  of  7  huts  is  128,  what  is  the  price  of 
1  hat  ? 

Analysis. — 1  is  1  seventh  of  7 ;  therefore  1  hat  will  cost  1  seventh  of 
$28,  and  1  seventh  of  $28  is  $4,  Ans. 

2.  If  1  man  can  hoe  a  field  of  corn  in  40  days,  how  long 
will  it  take  5  men  to  hoe  it  ? 

3.  A  grocer  bought  8  barrels  of  flour  for  $56  ;  what  must  he 
sell  it  for  per  barrel  to  gain  $12  ? 

4.  A  hardware  merchant  paid  169  for  7  kegs  of  nails,  and 
$15  freight ;  what  did  each  keg  cost  bim  ? 

5.  How  many  times  9  in  8  times  12? 

6.  How  many  times  11  in  74  plus  13  ? 

7.  How  many  cords  of  wood  at  $4  a  cord,   will  pay  for 
8  pairs  of  boots  at  16  a  pair  ? 

8.  A  farmer  gave  5  tons  of  hay  for  2  cows,  worth  $30  apiece ; 
what  was  the  value  of  the  liay  per  ton  ? 

9.  How  many  tons  of  coal,  worth  $6  a  ton,  must  I  give  for 
5  suits  of  clot  lies  worth  $9  a  suit  ? 

10.  If  I  pay  $12  apiece  for  7  barrels  of  beef,  and  sell  it  so  as 
to  lose  $24,  what  shaU  I  get  a  barrel  ? 

Written     Exercises. 

112.  Ex.  1.  Divide  172859  by  34. 

Explanation.— 34  is  contained  in  172,  operation. 

5  (thousands)  times  and  2  (thousands)  rem.  34  )  172859  (  5084-3^j 

Setting  the  5  at  the  right  of  the  dividend,  '[^Q 

we  annex  the  8,  the  next  figure  of  the  divi- 
dend,  to  the  remainder  2,  making  28  for  the  '^^'^ 

second  partial  dividend.  272 

As  33  is  not  contained  in  28,  we  write  a  -.  oq 

cipher  in  the  quotient,  and  annex  the  5  to 
28,  making  285  (tens).     Now  34  is  in  285  H!? 

(tens),  8   (tens)    times.      Writing   8   in   the  3 

quotient,  we   multiply,  subtract,   etc.,    and 

proceed  as  before.     Finally,  we  write  the  divisor  under  the  remainder, 
and  place  it  at  the  right  as  part  of  the  quotient. 


56 


Division. 


(2.) 
16  )  45807  ( 

(5.) 
19  )  560372  ( 

(8.) 
48  )  9230.56  ( 

(11.) 

47  )  159.85  ( 


(3.) 

28  )  348072  ( 

(6.) 

36  )  39245  ( 

(9.) 

29  )  702345  ( 

(12.) 

56  )  175.845  ( 


(4.) 

37  )  516780  ( 

(7.) 

37  )  45567  ( 

(10.) 

38  )  145.84  ( 

(13.) 
48  )  196.354  ( 


113.  From  the  preceding  illustrations  we  derive  the  following 


General     Rule. 

/.  Write  the  divisor  at  the  left  of  the  dividend,  and 
find  how  many  times  it  is  contained  in  the  feiuest 
orders  that  will  contain  it,  setting  the  quotient  at  the 
right. 

II.  Multiply  the  divisor  hy  this  quotient,  and  sub- 
tract the  product  from  the  orders  divided.  To  the 
r^inainder  anneoc  the  succeeding  figure  of  the  divi- 
dend, and  divide  as  before. 

III.  If  there  is  a  remainder  after  dividing  the  last 
order,  write  it  over  the  divisor,  and  place  the  result  at 
the  right  as  part  of  the  quotient. 

Finally ,  point  off  as  many  decimal  figures  at  the 
right  of  the  quotient  as  the  decimal  places  in  the  divi- 
dend exceed  those  in  the  divisor. 

Proof. — Multiply  the  divisor  and  quotient  together, 
and  to  the  product  add  the  remainder.  If  -the  result 
is  equal  to  the  dividend,  the  worh  is  right. 

Note. — The  quotient  figure  both  in  short  and  long  division,  is  always  the 
same  order  as  the  right  hand  order  divided. 

114.  To  prove  Multiplication  by  Division. 

Divide  the  product  by  one  of  the  factors,  and  if  the 
quotient  is  equal  to  the  other  factor,  the  worh  is  right 


Division.  57 


Appi^ications, 

115.     1.  A  man  wishes  to  invest  $2562  in  railroad  stock ; 
how  many  shares  can  he  buy,  at  $42  per  share  ? 

2.  In  1  year  there  are  52  weeks ;  how  many  years  are  there 
in  1640  weeks? 

3.  In  one  hogshead  there  are  63  gallons ;   how  many  hogs- 
heads are  there  in  3065  gallons? 

4.  If  a  man  can  earn  175  in  a  month,  how  many  months  will 
it  take  him  to  earn  -S3280  ? 

5.  If  it  takes  18  yards  of  silk  to  make  a  dress,  how  many 
dresses  can  be  made  from  1350  yards? 

6.  If  a  3'onng  man's  expenses  are  $83  a  month,  how  long 
will  $4265  support  him  ? 

7.  A  man  bought  a  drove  of  95  horses  for  $4750  ;  how  much 
did  he  give  apiece  ? 

8.  A  farmer  having  $1840,  laid  it  out  in  land,  at  $25  per 
acre ;  how  many  acres  did  he  buy  ? 

9.  In  a  cask  there  are  93  gallons  ;  how  many  casks  in  4260 
gallons  ? 

10.  If  a  man  travels  45  miles  a  day,  how  long  will  it  take 
him  to  travel  1215  miles  ? 

Divide  and  prove  the  following  : 


11. 

$467.2  by  15. 

19. 

$84.53  by  62. 

12. 

$56.84  by  18. 

20. 

$73.56  by  48, 

13. 

786.3  by  21. 

21. 

6893  by  82. 

14. 

48.27  by  33. 

22. 

9721  by  65. 

15. 

6972  by  35. 

23. 

23456  by  28. 

16. 

7842  by  23. 

24. 

72350  by  45. 

17. 

8253  by  47. 

25. 

80854  by  84. 

18. 

21.08  by  32. 

26. 

92635  by  92. 

27.  A  garrison  had  5580  pounds  of  beef,  which  the  com-« 
mander  wished  to  have  last  62  days  ;  how  many  pounds  could 
be  used  per  day  ? 

28.  A  man  paid  $9565  for  a  farm,  at  $64  an  acre;   how 
many  acres  were  there  ? 


58  Division, 

29.  A  grocer  packed  18144  eggs  in  boxes  holding  144  eggs 
each  ;  how  many  boxes  did  he  use  ? 

30.  If  he  had  packed  the  same  eggs  in  63  equal  boxes,  how 
many  eggs  would  he  have  put  in  a  box  ? 

Note. — When  the  divisor  is  large,  find  how  many  times  its  first  figure 
is  contained  in  the  first  or  first  two  figures  of  the  dividend,  allowing  for 
the  addition  of  tens  from  the  product  of  the  second  figure  of  the  divisor. 

31.  Divide  1814G  by  683.  A^is.   20||f. 

32.  62346  by  254.  40.  89256.48  by  732. 

33.  70893  by  532.  41.  2439.2642  by  765. 

34.  294763  by  306.  42.  592348.276  by  879. 

35.  375426  by  521.  43.  569389.175  by  1247. 

36.  2445224  by  812.  44.  8679538.46  by  3238. 

37.  3560325  by  904.  45.  134259.8640  by  56813. 

38.  4256348  by  638.  46.  396478.9523  by  75436. 

39.  5437502  by  743.  47.  425367.805  by  83247. 

Dictation     Exercises.* 

116.  1.  Subtract  4  from  11,  add  3,  multiply  by  6,  divide 
by  10,  add  6,  subtract  4,  multiply  by  5,  and  divide  by  8;  what 
is  the  result  ? 

2.  Multiply  9  by  7,  subtract  3,  divide  by  5,  add  10,  divide 
by  11,  multiply  by  8,  subtract  4,  and  add  12  ;  result  ? 

3.  Divide  54  by  9,  multiply  by  3,  subtract  8,  add  4,  divide 
by  7,  multiply  by  9,  add  10,  subtract  7,  divide  by  3  ;  result? 

4.  If  from  39  you  take  7,  divide  by  8,  multiply  by  9,  sub- 
tract 6,  divide  by  5,  add  12,  divide  by  9,  multiply  by  11,  add  3, 
divide  by  5,  add  7,  and  multiply  by  4,  what  is  the  result  ? 

5.  7  +  8  —  3x4^6  +  10 -^3x7  —  2;   result  ?  f 

6.  30  —  6-^8x9  +  5-^4  +  14 -f- 11  X  12 +  6-^5x  7 
+  6  -^  8  +  21  ;  what  is  the  result  ? 

7.  8x7  —  8-^4x2^6  + 16 -^4x8  —  12  -^7  +  6 
X  6  ;  what  is  the  result  ? 

*  The  object  of  these  exercises  is  three-fold  ;  First,  to  give  facility  in  mental  com- 
binations of  numbers ;  Second,  to  cultivate  the  habit  of  fixing  the  attention  ;  Third,  to 
drill  the  whole  class  at  the  same  time. 

t  Perform  the  successive  operations  indicated  by  the  signs. 


Division,  59 

8.  48  -7-  6  X  4  +  10  —  G  -M)  +  30  —  7  H-  3  X  12  —  5  +  7 
-f-  10  X  3  ;  what  is  the  result  ? 

9.  25  +  7->8xll  —  8^9  +  23-^9x7  +  12-^3  —  5 
X  8  +  6  -^  9  ;  what  is  the  result  ? 

10.  27  -^  9  +  15  —  10  X  7  +  7  -4-  9  X  5  —  8  -^  3  +  12 
_^_  7  X  20  —  12  -^  6  4-  25  =  how  many  ? 

117.  When  the  Divisor  has  Ciphers  on  the  right. 

1.  Divide  3563  by  100. 

Explanation.— Cutting  off  the  right-  operation. 

hand  figure  of  a  number,  removes  each  of  1|00  )  35|63 

its  other  figures  one  place  to  the  right,  i         q"    ftQ 

and   therefore  divides  it  bj  10.     Cutting 

off  two  figures  divides  it  by  100  ;  cutting  off  three  figures  divides  it  I  y 
1000,  etc.     (Art.  90.) 

2.  Divide  345231  by  100.    4.  Divide  6423544  by  10000. 

3.  Divide  672487  by  1000.    5.  Divide  7364159  by  100000. 

6.  Divide  937643  by  4000. 

Analysis. — By  cutting  off  three  figures  operation. 

at  the  right  of  the  divisor  and  dividend.  41000  )  937 1 643 

we  divide  each  by  1000;    the  quotient  is  .         TTTi    ^TTT^o 

no-r         ^  ^1  ■     ^        c^Ao        ^      *    a-      a  AllS.    234-1643       Rem. 

937  and  the  remainder  643.     Next,  divid- 
ing by  4,  the  quotient  is  234,  and  1  remainder,  which  we  prefix  to  the 
figures  cut  off,  making  the  true  remainder  1643.     Hence, 

118.  To  Divide  by  10,  100,  1000. 

Cut  off  as  many  figures  at  the  right  of  the  dividend 
as  there  are  ciphers  in  the  divisor;  the  remaining 
figures  will  he  the  quotient,  and  those  cut  off'  the 
remainder. 

When  the  divisor  is  greater  than  I,  with  ciphers  on  the  right. 

Cut  off  the  ciphers  from  the  divisor  and  as  many  fig- 
ures from  the  right  of  the  dividend. 

For  the  quotient,  divide  the  remaining  part  of  the 
dividend  by  the  remaining  part  of  the  divisor. 

To  the  figures  cut  off,  prefix  the  remainder ,  and  the 
result  will  be  the  true  remainder.     (Art.  870,  Appendix.) 


60  Divmon, 

7.  Divide  4885970  by  6000.  Ans.  814  and  1970  rem. 

8.  Allowing  200  lbs.  to  a  barrel,  how  many  barrels  will 
68000  lbs.  of  beef  make  .^ 

9.  In  $1  there  are  100  cents  ;  how  many  dollars  are  in 
45650  cents  ? 

10.  How  many  bales  of  cotton,  each  weighing  450  lbs.,  are 
in  36000  lbs.  ? 

11.  If  $96000  are  divided  equally  among  2400  soldiers,  how 
much  will  each  receive  ? 

12.  A  pound  of  cotton  has  been  spun  into  a  thread  76  miles 
long,  and  a  pound  of  wool  into  a  thread  95  miles  long ;  how 
many  pounds  of  both  together  will  spin  a  thread  which  will 
reach  round  the  world,  a  distance  of  25000  miles  ? 

13.  If  600  steam  engines  can  do  the  work  of  2  million 
496  thousand  men,  to  how  many  men  is  1  engine  equivalent  ? 

119.  From  the  relations  of  the  Divisor,  Divide^id,  and 
Quotient,  we  deduce  the  following 

General     Principles    of     Division, 

First. — Let  24  be  a  dividend  and  6  a  divisor.  The  quotient 
is  4. 

Then  (24  X  2)  -  6  =  8  I  _ 

And    24  --  (6  -^  2)  =  8  S  -  *  ^  ''•     ^^"^^^^ 

i°.  Mnltiplying  the  dividend,  or 
Dividing  the  divisor, 

8econd.-{U  -^  2)  -  6  =  2  |  _ 

And  24  --  (6  X  2)  =  2  (  -  ^   *   ^-     ^''''^^' 


[  Multiplies  the  quotient. 


2°.  Dividinq    the    dividend,  or     ^.   .,     ,,  ,.     , 

,^  ,  .  y  .       ,-      ,.  .  \  Divides  ihQ  (moiiQui. 

Multiplying  the  divisor,         ) 

raM-rf.-(24  X  2)  -  (0  X  2)  ( 

Or,  (24  -^  2)  -^  (6  -  2)  f  -  *•     ^'^"°*' 

5°.  Multiplyinq  or  dividinq  both  )  ^  ,     .  ,, 

^.  :  ^    "^  -,    ,.  .,      ,  ,     (  Does  «o/  change  the  quo- 

di visor   and   dividend  by  /■       . 

the  same  number,  ' 


Division.  61 

119,  a,  1st.  When  the  jirodiict  of  two  factors  and  one  of 
them  are  given,  the  other  is  found  by  dividing  the  product  by 
the  given  factor. 

2d.  When  the  product  of  three  or  more  factors  and  all  but 
one  of  them  are  given,  the  other  factor  is  found  by  dividing 
the  given  product  by  the  product  of  the  given  factors. 

3d.  When  the  sum  and  difference  of  two  numbers  are  given, 
the  less  number  is  found  by  subtracting  the  difference  from  the 
sum  and  dividing  the  remainder  by  2. 

4th.  The  average  of  two  unequal  numbers  is  half  their  sum. 
The  average  of  three  unequal  numbers  is  one-third  the  sum. 

Oral    Problems    for    Review. 

120.  1.  The  dividend  is  63,  the  quotient  9;  what  is  the 
divisor  ? 

2.  When  the  dividend  and  quotient  are  given,  how  ^nd  the 
divisor  ? 

3.  The  divisor  being  11  and  the  quotient  10,  what  is  the 
dividend  ? 

4.  When  the  divisor  and  quotient  are  given,  how  find  the 
dividend  ? 

5.  The  quotient  being  9,  the  divisor  20,  and  the  remainder  7, 
what  is  the  dividend  ? 

6.  When  the  divisor,  quotient,  and  remainder  are  given,  how 
find  the  dividend  ? 

7.  At  17  a  week,  how  many  weeks  can  you  board  for  $84  ? 

8.  How  long  will  it  take  a  printer  to  earn  $132,  if  he  gets 
$11  a  week? 

9.  A  farmer  bought  12  yards  of  cloth,  at  $4,  and  paid  for  it 
in  hay,  at  $8  a  ton;  how  many  tons  did  it  take? 

10.  In  7  times  11,  less  5,  how  many  times  9  ? 

11.  In  9  times  12,  less  8,  how  many  times  5  ? 

12.  How  many  tons  of  coal,  at  $6  a  ton,  will  pay  for  8  bar- 
rels of  flour,  at  $9  a  barrel  ? 

13.  If  12  men  can  earn  $100  in  a  week,  how  much  can  1  man 
earn  in  the  same  time  ? 


62  Division. 

14.  When  wood  is  14  a  cord  and  coal  is  $9  a  ton,  how  much 
wood  is  equal  in  value  to  8  tons  of  coal  ? 

15.  If  eggs  are  worth  9  cents  a  dozen,  and  butter  12  cents  a 
pound,  how  many  eggs  are  worth  6  lb.  of  butter? 

16.  A  man  being  on  a  journey,  finds  he  can  reach  home  in 
9  days  by  traveling  20  miles  a  day;  but  becoming  lame,  he 
traveled  onlv  12  miles  a  day ;  in  how  many  days  did  he  reach 
home  ? 

17.  A  man  bought  6  hats  at  |4  apiece,  and  5  caps  at  12,  and 
paid  in  apples  at  16  a  barrel ;  how  many  barrels  and  what  part 
of  a  barrel  did  it  take  to  pay  the  bill  ? 

18.  John  has  12  marbles  and  William  has  9  times  as  many  as 
John,  minus  11  ;  how  many  marbles  has  William? 

19.  When  peaches  are  sold  at  the  rate  of  5  for  8  cents,  how 
many  will  56  cents  buy  ? 

20.  What  cost  60  apples,  at  the  rate  of  10  for  7  cents  ? 

21.  George  bought  12  oranges,  at  4  cents  apiece,  and  after 
eating  3  of  them,  sold  the  rest  at  6  cents  apiece ;  did  he  make 
or  lose  by  his  bargain,  and  how  much  ? 

Written    Problems    for    Review. 

121.  1.  The  product  of  two  numbers  being  252,  and  the 
multiplier  18,  what  is  the  multiplicand? 

2.  The  product  of  two  numbers  is  576,  the  multiplicand  48 ; 
what  is  the  multiplier? 

3.  When  the  product  of  two  factors  and  one  of  the  factors 
are  given,  how  find  the  other  factor  ? 

4.  The  sum  of  two  numbers  is  250,  their  difference  50  ;  what 
is  the  smaller  number  ?     The  greater  ? 

5.  At  an  election  A  and  B  together  received  273  votes,  and 
A  had  37  more  than  B ;  how  many  had  each  ? 

6.  A  grocer  mixed  two  kinds  of  tea  in  equal  quantities, 
worth  63  and  75  cts.  a  pound  respectively;  what  is  the  average 
price  of  the  mixture  a  pound  ? 

7.  What  is  the  average  age  of  3  brothers,  who  are  respectively 
76,  81,  and  89  years  old? 


Division.  63 

8.  What  is  the  average  price  of  4  horses,  worth  respectively 
$180,  $273,  $804,  and  $375  ? 

9.  The  ship  America  of  Boston,  sailed  56  hours  at  the  rate 
of  11  miles  per  hour,  when  she  encountered  a  storm  of  10 
hours  duration  which  drove  her  back  at  the  rate  of  14  miles 
per  hour  ;  how  far  from  port  was  she  at  the  end  of  72  hours? 

10.  A  thief  fled  from  New  York,  at  the  rate  of  85  miles  a 
day ;  5  days  after  an  officer  started  in  pursuit  of  him  at  the 
rate  of  138  miles  a  day  ;  how  far  from  the  thief  was  the  officer 
at  the  end  of  8  days  from  the  time  the  latter  started  ? 

11.  A  is  worth  $1265,  B  is  worth  4  times  as  much  as  A,  and 
$183,  and  C  is  worth  three  times  as  much  as  A  and  B  lacking 
$2348 ;  how  much  are  B  and  C  worth  respectively ;  and  how 
much  are  they  all  worth  ? 

12.  If  a  man's  salary  is  $3176  a  year,  and  he  spends  $7  a 
day,  how  much  can  he  lay  up  ? 

13.  In  a  single  city,  $2170  are  spent  daily  for  cigars ;  how 
many  free  schools  will  this  support,  at  $1085  each  per  annum  ? 

14.  A  man  bought  467  acres  of  land,  at  $16  per  acre,  and 
sold  it  for  $9340  ;  how  much  did  he  get  per  acre  ;  and  how 
much  did  he  gain  or  lose  by  his  bargain  ? 

15.  A  man  bought  563  horses,  at  $65  apiece,  and  sold 
them  so  as  to  make  $860  ;  how  much  did  he  get  apiece  ? 

16.  Which  are  worth  more,  863  cows  at  $38  apiece,  or  356 
horses  at  $75  apiece  ?    How  much  ? 

17.  A  owns  1368  acres  of  wild  land,  which  is  6  times  as 
much  as  B  owns,  and  B  owns  twice  as  much  as  C  ;  how  much 
land  do  B  and  C  own ;  and  how  much  do  all  own  ? 

18.  The  smaller  of  two  numbers  is  contained  14  times  in  252, 
the  greater  is  49  times  the  smaller  ;  what  are  the  numbers  ? 

19.  A  man  bought  a  drove  of  oxen  for  $18130,  and  after 
selling  84  of  them  at  $51  apiece,  the  rest  stood  him  in  $43 
apiece  ;  how  many  did  he  buy? 

20.  What  is  the  difference  between  9313702853  divided  by 
1987,  and  46481  multiplied  by  936? 

21.  A  man  sold  155  acres  of  land  at  $34  per  acre,  and  took 
in  payment  for  it,  19  horses  at  $65  apiece,  and  15  cows  at  $17 
apiece  ;  how  much  was  still  due  him  ? 


A. 


64  Division, 

22.  AYhat  number  besides  137  will  exactly  divide  11371  ? 

23.  The  quotient  being  275,  the  divisor  383,  and  the  remain- 
der 49,  what  is  the  dividend  ? 

24.  If  the  dividend  is  2756,  the  quotient  184,  and  the  re- 
mainder 180,  what  is  the  divisor? 

25.  What  must  5376  be  multiplied  by,  to  make  6521088  ? 

26.  How  many  times  can  437  be  subtracted  from  18791  ? 

27.  If  the  sum  of  14350  and  7845  is  divided  by  965,  the 
quotient  multiplied  by  386,  and  the  product  diminished  by  761, 
what  will  the  remainder  be? 

28.  The  sum  of  250  and  173,  being  multiplied  by  their  differ- 
ence, and  the  product  divided  by  45,  what  is  the  quotient  ? 

29.  How  many  men  will  it  take  to  do  as  much  work  in  1 
day,  as  368  men  can  do  in  134  clays  ? 

30.  How  many  men  would  be  required  to  do  the  same  work 
in  16  days? 

31.  Four  men.  A,  B,  0,  and  D,  bought  a  ship  together  for 
116256  ;  A  paid  14756,  B  paid  1763  more  than  A,  and  C  $256 
less  than  B  ;  how  much  did  D  pay  ? 

32.  Bought  sofas  for  19212  and  selling  them  at  167  gained 
120  on  each ;  how  many  were  bought  ? 

Q  U  ESTI  O  N  S. 

95.  What  is  Division?  96.  The  Dividend?  97.  Divisor?  98.  What 
is  the  answer  called  ?     What  does  it  show  ?     99.  Remainder  ? 

100.  Make  the  sign  of  division.  How  is  it  read  ?  101.  How  else  is 
division  denoted  ? 

102.  When  a  number  is  divided  into  two  equal  parts,  what  is  one  of 
the  parts  called?  104.  When  a  unit  is  divided  into  equal  parts,  what  are 
the  parts  called  ? 

106.  When  the  divisor  and  dividend  are  like  numbers,  what  is  the 
quotient  ?  When  the  divisor  is  an  abstract  number,  \vhat  are  the  dividend 
and  quotient  ?     To  what  is  the  product  of  the  divisor  and  quotient  equal  ? 

110.  How  divide  when  the  dividend  has  decimals?  113.  What  is  the 
general  rule  ?  How  prove  division  ?  118.  How  divide  by  10,  100, 
1000,  etc.  ?  When  the  divisor  is  greater  than  1,  with  ciphers  on  the  right, 
how  proceed  ? 

119.  What  is  the  effect  of  multiplying  the  dividend  or  dividing  the 
divisor?  Of  dividing  the  dividend  or  multiplying  the  divisor?  Of  multi- 
plying or  dividing  both  by  the  same  numl^er? 


them  thus,  --^ — ^—  ;  what  factors  are  common  to  both  ? 


^CANCELLATION. 

Devjsloi^ment     of     PbINCIPIjES. 

122.  1.   What  is  the  quotient  of  24  divided  by  6  ?     Ans.  4. 

2.  Separate  the  dividend  and  divisor  into  factors,  and  write 
2x3x4 

2x3 

3.  If  you  cancel  the  factor  2,  w^hich  is  common  to  both,  what 
is  the  quotient  ?     A  ns.  4. 

Note. — To  cancel  means  to  cross  out  or  reject. 

4.  If  you  cancel  both  the  2's  and  the  3's,  what  is  the  effect  ? 
Ans.  The  quotient  is  not  altered.    Hence,  the  following 

Principles. 

123.  i°.   CancelU7ig  a  factor  of  a  number  divides  the  nuniber 
hy  that  factor. 

2°,   Cancelling  equal  factors  of  the  divisor  and  dividend  does 
not  change  the  quotient.     (Art.  119,  5°.) 

124.  Cancellation  is  the  method  of  shortening  Division,  by 
rejecting  equal  factors  from  the  divisor  and  dividend. 

The  Sign  of  Cancellation  is  an  oblique  mark  drawn  across 
the  face  of  a  figure  ;  as,  $,  ^,  ^,  etc. 

125.  To  divide  by  Cancellation. 

5.  Divide  the  product  of  14  x  15  x  56  by  8  x  45  x  7. 

1st  form.  2d  FOEItt. 

14  =  4f,  Ans. 
Explanation. — Since  8  in  the  divisor  is  a  factor  of  56  in  the  dividend. 


2  1  P 

$x0xti    -     3     -^^'  ^'''*  .     * 

3  

3 


66  Cancellation. 

cancel  the  8  in  both,  retaining  7,  the  other  factor  of  56.  Also  cancel 
15,  a  factor  of  45,  and  7  a  factor  of  14,  retaining  the  prime  factor  2  in 
the  dividend,  and  3  in  the  divisor ;  then  (7  x  3)  -r-  3  =  4|,  Ans.     Hence,  the 

EuLE. — Cancel  all  the  factors  comnioii  to  the  divisor 
and  dividend,  and  divide  the  product  of  those  remain- 
ing in  the  dividend  by  the  product  of  those  remaining 
in  the  divisor.     (Art.  123,  2°.) 

Note. — When  a  factor  cancelled  is  equal  to  the  number  itself,  the  unit 
1  always  remains.  If  the  1  is  in  the  dimdend  it  must  be  retained  ;  if  in 
the  divisor,  it  may  be  disregarded. 

What  is  the  quotient  of 

6.  28x56x15-^14x5x3?  10.     1365-^21x5? 

7.  112x40x18-^56x3x4?         11.     2850-^125? 

8.  48x72x20-^48x15x7?         12.     3236-^256? 

9.  54x36x25-f-45x7x30?         13.     1728-^576  ? 
14.  120  X  24  X  35  X  9-f-42  x  15  x  54  x  7  ? 

15.  An  agent  sold  176  boxes  of  starch,  of  15  lbs.  each,  at 
12  cts.;  how  many  loads  of  corn,  having  9  sacks  of  5  bu.  each, 
worth  44  cts.  a  bushel,  will  it  require  to  pay  for  the  starch  ? 

16       "a       4 

The  val.  of  starch  =  176x  15x12)       ^lUxUxU     ,.    , 

^      ,,  >  and  — - — ^ — -—:=lh,An§. 
''       ''       corn    =     9x5x44j  ^x$xM 

~a  lis 

Note. — Practical  Problems,  should  first  be  analyzed,  and  the  o[3erations 
indicated.     Then  cancel  as  before. 

16.  A  farmer  bought  9  cows  at  125  apiece,  and  paid  for  them 
in  hay  at  115  a  ton  ;  how  many  tons  of  hay  did  it  require  ? 

17.  How  many  bags  of  coffee  containing  5G  lbs.,  at  28  cts.  a 
pound,  must  be  given  for  8  pieces  of  muslin,  each  containing 
40  yards,  at  8  cts.  a  yard  ? 

18.  How  many  barrels  of  flour  worth  18  a  barrel,  must  be 
given  for  45  tons  of  coal  at  |6  a  ton  ? 

19.  A  miller  bought  7  loads  of  wheat,  each  containing  28  bags 
of  3  bushels  each,  worth  $1.50  a  bushel,  and  paid  for  it  m  flour 
at  $7  a  barrel ;  how  much  flour  was  required  ? 


* 


(tJ    ■v'.S^   -^ 


EOPEETIES     OF 


\D  ^•'isiv  ^ 


Definitions. 

126.  Numbers  are  diyided  into  Odd,  Even,  Prime,  and 
Composite. 

127.  An  Even  Number  is  one  that  can  be  exactly  divided  by  2. 

128.  An  Odd  Number  is  one  that  cannot  be  exactly  divided 
by  2 ;  as  3,  5,  1,  etc. 

129.  A  Prime  Number  is  one  that  cannot  be  exactly  divided 

by  any  number,  except  a  unit  and  itself;  as  5,  7,  11,  etc. 
Note. — All  prime  numbers  except  2  are  odd. 

130.  Two  numbers  are  Prime  to  each  other  when  the  only 
number  by  which  both  can  be  exactly  divided  is  a  unit  or  one ; 
as  5  and  6. 

131.  A  Composite  Number  is  the  product  of  two  or  more 
factors,  each  of  which  is  greater  than  1 ;  as  21  =  3  x  7. 

Note. — The  least  divisor  of  a  Composite  Number  is  a  prime  number. 

132.  An  Exact  Divisor  of  a  number  is  one  which  will  divide 
it  without  a  remainder. 

One  number  is  said  to  be  divisiUe  by  another  when  there 
is  no  remainder. 

133.  The  Factors  of  a  number  are  the  numbers  whose 
product  equals  that  number.     (Art.  80.) 

Thus,  7  and  9  are  the  factors  of  63  ;  8,  4  and  5,  of  60. 

134.  A  Prime  Factor  is  a  prime  number  used  as  a  factor. 
Note. — The  prime  factors  of  a  number  are  also  exact  divisors  of  it. 

135.  The  Reciprocal  of  a  number  is  1  divided  by  that 
number.    Thus,  the  reciprocal  of  4  is  1  -^  4,  or  \. 


68  Properties  of  Numhers, 

Oral     Ex  e  rcises. 

136.  1.  Name  the  eyen  numbers  up  to  31. 

2.  Name  the  odd  numbers  less  than  30. 

3.  Name  the  prime  numbers  less  thin  30. 

4.  Name  the  composite  numbers  up  to  30. 

5.  Name  an  exact  divisor  of  18,  27,  42. 

6.  Name  all  the  exact  divisors  of  24  ;  of  36. 

7.  What  is  the  smallest  number  except  1,  that  will  exactly 
divide  10?     15?     25?    35?     49? 

8.  What  is  the  largest  number,  except  itself,  that  will  exactly 
divide  18?    22?     24?     30?     36? 

9.  AVhat  numbers  multiplied  together  j^roduce  21  ?  35  ? 
42?     27?     45?     48? 

10.  What  will  produce  33  ?    54  ?    63  ?     36  ?     72  ? 

DeVEIjOPMENT    of    JPltlNCirLES. 

137.  First. — Take  any  number,  as  12,  and  separate  it  into  the 
factors  3  and  4. 

If  we  multiply  12  by  2  the  product  is  24,  if  we  multiply  it 
by  3  the  product  is  36,  etc.  Now  each  of  these  products  is 
divisible  by  3  and  by  4.     Hence, 

i°.  If  one  7imy}her  is  a  factor  of  another,  the  former  is  also  a 
factor  of  any  Product  or  Multiple  of  the  latter. 

Second. — Take  any  number,  as  2,  which  is  a  common  factor 
of  4  and  12. 

The  sum  of  4  +  12  =  16;  their  difference  12— 4  =r  8,  and 
their  product  12  x  4  =  48.  By  inspection  we  see  that  2  is  a 
factor  of  16,  of  8,  and  of  48.     Hence, 

^°.  A  factor  common  to  two  or  more  numhers,  is  also  a  factor 
of  their  Sum,  their  Difference,  and  their  I*roduct. 

Third. — Take  any  composite  number,  as  30. 

30  is  divisible  by  2,  3  and  5  ;  also  by  2  x  3,  or  6  ;  by  2  x  5, 
or  10  ;  by  3  X  5,  or  15  ;  and  by  no  other  number.  But  2,  3 
and  5  are  its  prime  factors ;  6,  10  and  15  are  the  different 
products  of  them.     Hence, 

«5°.  Every  composite  number  is  divisible  by  each  of  its  Prime 
factors;  and  by  the  Product  of  any  ttvo  or  more  of  them. 


<1^„     ■ 


I 


■      )l(      .  (g 1* 

ACTORING. 


Definitions. 


138.  Factoring  a  number  is  separating  it  into  factors. 
Thus,  the  factors  of  21  are  3  and  7  ;  the  factors  of  32  are  4  and  8. 

139.  A  Composite  Number  is  separated  into  ttvo  factors  by- 
dividing  it  by  any  exact  divisor. 

Note. — It  is  not  customary  to  consider  the  unit  1  and  the  number 
itself  as  factors  ;  if  they  were,  all  numbers  would  be  composite.     (Art.  131.) 

140.  A  number  that  is  a  factor  or  divisor  of  two  or  more 
numbers  is  called  a  Common  Factor  or  Common  Measure  of 
those  numbers. 

141.  The  following  facts  will  assist  the  learner  in  separating 
large  numbers  into  factors  : 

All  numbers  are  divisible 

i°.  By  2,  which  end  with  a  cipher,  or  a  digit  divisible  by  2. 

2°.  By  3,  when  the  sum  of  the  digits  is  divisible  by  3. 

3°.  By  4,  when  the  number  expressed  by  the  two  right  hand 
figures  is  divisible  by  4. 

Jf°.  By  5,  which  end  with  a  cipher  or  5. 

5°.  By  6,  when  divisible  by  2  and  3. 

6°.  By  8,  W'hen  the  three  right  hand  figures  are  ciphers,  or 
when  the  number  expressed  by  them  is  divisible  by  8. 

7°.  By  9,  when  the  sum  of  the  digits  is  divisible  by  9. 
(See  Art.  875,  Appendix.) 

^°.  By  10,  100  or  1000,  which  end  with  an  equal  number 
of  ciphers. 

Note. — For  7,  no  convenient  rule  can  be  given 


70 


Properties  of  Numhers, 


2 

2310 

Given  Namber. 

3 

1155 

1st  Quotient. 

5 

385 

2d  Quotient. 

7 

77 

3d  Quotient. 

11 

4th  Quotient. 

Oral     Exercises. 

142.  1.  What  prime  factors  will  exactly  divide  12  ?  18  ?  26  ? 

2.  What  prime  factors  will  exactly  divide  30  ?     36  ?     40  ? 

3.  What  prime  factors  are  common  to  18,  24,  and  36  ? 

4.  Name  the  prime  factors  common  to  45,  27,  and  60  ? 

Written    Exercises. 

143.  To  Separate  a  Number  into  Prime  Factors. 

1.  What  are  the  prime  factors  of  2310  ? 

Explanation. — We  divide  tlie  given 
number  by  any  prime  factor,  as  2,  and  the 
successive  quotients  by  the  prime  factors 
3,  5  and  7,  and  the  last  quotient  11,  is  a 
prime  number.  Therefore,  the  several 
divisors  with  the  last  quotient  are  the  prime 
factors  required. 

Proof.— 2  x  3  x  5  x  7  x  11=2310.     Hence,  the 

Rule.  —  Divide  the  given  munhev  hy  any  prime 
facto?' ;  then  divide  this  quotient  by  another  prime 
factor ;  and  so  on  until  the  quotient  obtained  is  a  prime 
number.  The  several  divisors,  with  the  last  quotient, 
are  the  prime  factors  required. 

Find  the  prime  factors  of 

2.  225.  e.  672.  10.  3420.  14.  10376. 

3.  376.  7.  796.  11.  18500.  15.  25600. 

4.  344.  8.  864.  12.  46096.  16.  64384. 

5.  576.  9.  945.  13.  96464.  17.  98816. 

144.  To  find  the  Prime  Factors  common  to  two  or  more  numbers. 

18.  Find  the  prime  factors  common  to  168,  42,  and  210  ? 

2  )  168,     42,     210 

3  )    84,     21,     105 
7  )    28,       7,      35 

4,       1,        5 


Explanation.  —  Dividing  by  the  prime 
factor  2,  the  quotients  are  84,  21,  and  105. 
Dividing  these  by  3,  we  have  28,  7,  and  35. 
Dividing  by  7,  the  quotients  are  prime  to  each 
other  (Art.  130).  The  divisors  2,  3,  and  7,  are 
the  prime  factors  required.     Hence,  the 


FactoriiKj.  71 

EuLE. — Divide  the  given  ninnhers  hy  any  common 
prime  factor,  and  the  quotients  thence  arising  in  like 
manner,  till  they  have  no  common  factor ;  the  several 
divisors  will  he  the  primps  factors  required. 

19.  What  are  the  prime  factors  common  to  24,  76,  and  32  ? 

20.  28,  54,  and  48  ?  24.     120,  96,  and  384  ? 

21.  58,  64,  and  84  ?  25.     168,  320,  and  256  ? 

22.  436,  308,  and  506  ?  26.     225,  350,  and  475  ? 

23.  252,  126,  and  210  ?  27.     144,  276,  and  524  ? 

Common    Divisors. 

145.  A  Common  Divisor  is  one  that  will  divide  ttvo  or  more 
numbers  without  a  remainder.  It  is  often  called  a  Common 
Measure. 

146.  The  Greatest  Common  Divisor  or  Greatest  Common 
Measure  of  two  or  more  numbers  is  the  greatest  number  that 
w^ill  divide  each  of  them  without  a  remainder,* 

Thus,  the  greatest  common  divisor  of  18  and  30  is  6. 
Note. — Numbers  which  are  prime  to   each  other  have   no  common 
divisor  or  measure  greater  than  1. 

Oral     Exercises. 

147.  1.  What  divisor  is  common  to  15  and  27  ? 

2.  What  divisor  is  common  to  16  and  20  ? 

3.  Find  a  common  factor  of  15,  18,  and  24. 

4.  What  is  the  greatest  number  that  will  divide  21  and  35 
without  a  remainder  ? 

5.  What  is  the  greatest  divisor  common  to  30  and  48  ?  Hence, 

148.  The  greatest  common  divisor  of  two  or  more  numbers 
is  the  iwoduct  of  all  their  common  inime  factors. 

Illustration.— Take  any  two  numbers,  as  30  and  42,  and  separate 
them  into  their  prime  factors  ;  thus, 

30  =  2  X  3  X  5  ;        42  =  2  x  3  x  7. 
Now  2  and  3  are  the  only  prime  factors  common  to  both  numbers,  and 
their  product,  6,  is  the  greatest  di^^sor  common  to  both. 

*  The  letters  g.  e.d.  etand  for  Greatest  Common  Divisor. 


72  Properties  of  Niimhers. 


Written     Exercises, 

149.  To  find  the  f/.  c,  d,   of  two  or   more   numbers  by  Prime 

Factors. 

1.  What  is  the  f/.  c,  d.  of  45,  30,  and  105  ? 

3  )  45,     30,     105         Or,     45  =  5  x  3  x  3 

5  )  15,     10, 35  30  =  5  X  3  X  2 

3,       2,         7  105  ==  5  X  3  X  7 

5  X  3  =  15,  </.  c.  f?.,  Ans. 

Explanation. — Separating  the  numbers  into  their  prime  factors,  we 
find  5  aud  3  common  to  each  ;  therefore  their  product  is  the  y,  e,  d, 
required.     (Art.   134.)     Hence,  the 

EuLE. — Separate  the  niniibers  into  their  prime  factors  ; 
the  product  of  those  that  are  common  to  each  is  the 
greatest  common  divisor. 

2.  Find  the  ff,  c,  d,  of  63  and  147. 

3.  91  and  117.  7.     16,  124,  and  300. 

4.  247  and  323.  8.     492,  744;  and  1044. 

5.  285  and  465.  9.     485,  145,  and  3471. 

6.  63,  105,  and  240.  10.     6430  and  8945. 

150.  To  find  the  g,  c.  d.  by  Continued  Division. 

1.  What  is  the  greatest  common  divisor  of  30  and  42  ? 

Analysis. — Dividing  the  greater  by  the  less,       30  )  42  (  1 
the   quotient   is   1    and   12   remainder.      Next,  oa 

dividing  the  first  divisor  by  the  first  remainder,  ■ — 

tlie  quotient   is  3   and   6   remainder.      Again,  12  )  30  (  2 

dividing  the  second  dinisor  by  the  second  re-  24 

mainder,  the   quotient  is  2  and  no  remainder.  ~  \  1  o  /  o 

Therefore,   6  is  the  </.   c.   d.   of    80  and   42.  6  )  12  (  2 

Hence,  the  12 

Rule. — Divide  the  greater  number  by  the  less ;  then 
divide  the  first  divisor  by  the  first  remainder,  and  so 
on,  until  nothing  remains ;  the  last  divisor  will  be  the 
greatest  common  divisor. 


Common  Divisors,  73 

If  there  are  more  thcni  tiro  niuiibers,  firid  the  greatest 
common  divisor  of  two  of  them ;  then  of  this  divisor 
and  a  third  number,  and  so  on,  until  all  the  numbers 
have  been  taken. 

Note. — The  greatest  common  divisor  of  two  or  more  prime  numbers, 
or  numbers  prime  to  each  other  is  1.     (Art.  130.) 

(For  demoustration,  see  Art.  873,  Aj^pendix.) 

1.  Find  the  r/.  c.  d,  of  246  and  324. 

2.  285  and  465.  6.     638296  and  33888. 

3.  72,  96,  and  132.  7.     18996  and  29932. 

4.  2145  and  3472.  8.     54128  and  262424. 

5.  464320  and  18945.  9.     143168  and  2064888. 

10.  A  farmer  has  664  bushels  of  oats  and  316  bushels  of  corn, 
which  he  wishes  to  send  to  market  in  the  largest  possible  bags 
of  equal  size  that  will  hold  each  kind  of  grain  ;  how  many 
bushels  must  each  bag  hold  ? 

11.  A  man  bought  three  pieces  of  land  containing  28,  36, 
and  44  acres  respectively,  which  he  wished  to  fence  into  the 
largest  possible  fields,  each  having  the  same  number  of  acres  \ 
how  many  acres  can  he  put  in  a  field  ? 

12.  A  grocer  had  42  oranges  and  63  pears  which  he  wished 
to  put  in  bags  each  containing  the  largest  number  i^ossible ; 
how  many  could  he  put  in  each  bag  ? 

13.  A  man  having  3  plots  of  land  fronting  a  street,  the 
width  of  which  was  600  ft.,  120  ft.,  and  900  ft.,  respectively, 
wished  to  divide  each  into  house-lots  of  equal  width ;  how  wide 
w^ill  the  lots  be,  and  how  many  can  be  made  from  each  plot  ? 

14.  Three  men  having  $1260,  $2268,  and  $2772  respectively, 
agreed  to  buy  horses  at  the  highest  price  per  head  that  will 
allow  each  man  to  invest  all  his  money  ;  how  rp^ny  horses  can 
each  man  buy  ? 


74  Properties  of  Numbers, 


Common    Multiples. 

151.  A  Multiple  of  a  number  is  one  which  is  exactly  divisi- 
ble by  that  number. 

Thus,  12  is  a  multiple  of  4  ;  18  of  6. 

152.  A  Common  Multiple  of  two  or  more  numbers  is  a  num- 
ber that  is  exactly  divisible  by  each  of  them. 

Thus,  18  is  a  common  multiple  of  2,  3,  6,  and  9. 

153.  The  Least  Common  Multiple  of  two  or  more  numbers, 
is  the  least  number  exactly  divisible  by  each  of  them.* 

Tims,  15  is  the  least  common  multiple  of  3  and  5, 

154.  A  composite  number  contains  all  the  prime  factors  of 
each  of  the  numbers  w^hich  produce  it. 

Development    of    I^rikcipzes. 

155.  1.  Name  two  numbers  each  of  which  can  be  divided 
by  3  and  5  without  a  remainder. 

2.  What  is  the  smallest  number  that  can  be  exactly  divided 
by  3  and  5  ? 

3.  Name  two  numbers  which    can  be   exactly  divided  by 
6  and  8  ? 

4.  What  is  the  smallest  number  that  can  be  exactly  divided 
by  6  and  8  ? 

5.  By  what  two  prime  numbers  can  35  be  divided  ? 

6.  What  is  the  least  number  that  is  exactly  divisible  by 

2,  3,  and  5  ? 

7.  What  is  the  least  number  that  can  be  exactly  divided  by 

3,  5,  and  6  ?     Hence,  we  derive  the  following 

Principles. 

156.  1°.  A  7nuUipIe  of  a  numher  must  contain  all  the  2)rime 
factors  of  that  number. 

3°.  A  common  mtiltiple  of  tioo  or  more  nnmhers  must  contain 
all  the  prime  factors  of  each  of  the  given  numbers. 

*  The  letters  I.  c.  in.  staucl  for  "  Least  Common  Multiple." 


Common  Multiples,  75 

3°.  The  least  common  multijjle  of  two  or  more  numiers  is  the 
least  numher  which  contains  all  their  prime  factors,  each  factor 
being  taken  the  greatest  numher  of  times  it  occurs  in  either  of 
the  given  numbers. 

Written     Exercises. 

157.  To  find  the  Least  Common  Multiple  of  two  or  more  numbers. 

1.  What  is  the  ?.  c.  m,  of  10,  21,  66  ? 

Explanation. — Write  the  numbers  in  a  line,  opekation. 

and  divide  them  by  any  prime  number  as   2,  2  )  10,      21,      66 

that  will  exactly  divide  two  or  more  of  them,  „  \     ^       ^^j        ^ 

settinor  the  quotients  and  undivided  numbers  in  '        '  '     1. 

a  line  below.     Divide  these  by  the  prime  number  5,        7,      11 

3,  and  set  the  results  below  as  before.      Now,  2x3x5x7x11 

as  all  the  numbers  in  the  third  line  are  prime,  oqin      A 

we  can  carry  the  division  no  further.     (Art.  129.)  ' 

Hence,  the  divisors  2  and  3,  with  tbe  numbers  in  the  last  line,  5,  7,  and 
11,  are  all  prime  factors  of  the  given  numbers,  and  each  is  taken  as  many 
times  as  it  occurs  in  either  of  them.  Therefore,  the  continued  product  of 
these  factors,  or  2310,  is  the  I,  c.  m.  required.    (Art.  156,  2^.)   Hence, 

158.  Rule.  —  JVHte  tJie  nurnhers  in  a  line,  and  divide 
by  any  prime  numher  that  will  divide  two  or  more  of 
them  without  a  remainder,  placing  the  quotients  cuid 
undivided  numbers  in  a  line  below. 

JVejot,  divide  this  line  as  before,  and'  thus  proceed  till 
no  two  numbers  are  divisihle  by  any  number  greater 
than  1.  Tlie  continued  product  of  the  divisors  and 
numbers  in   the  last   line  ivill    be   the   answer. 

Notes. — 1.  The  operation  may  often  be  shortened  by  cancelling  any 
number  which  is  a  factor  of  another  number  in  the  same  line. 

2.  When  the  given  numbers  are  prime  or  prime  to  each  other,  their  con- 
tinued product  will  be  the  least   common   multiple. 

2.  Find  the  I,  c,  m.  of  24,  16,  15,  and  20. 

3.  25,  60,  72,  and  35.       7.     17,  29,  53,  and  85. 

4.  63,  12,  84,  and  72.       8.     18,  55,  49,  33,  and  121. 

5.  54,  81,  14,  and  63.       9.     720,  336,  and  1736. 

6.  12,  72,  36,  and  144.    10.     8,  12,  16,  24,  36,  48,  72,  144. 


76  Common  Multvples. 

11.  Find  the  least  common  multiple  of  the  nine  digits. 

12.  Of  720,  336,  576,  and  1820. 

13.  Of  642,  876,  984,  and  2000. 

14.  Eequired  the  smallest  number  of  pears  that  a  farmer 
can  exactly  divide  among  3  classes  of  children  containing  18, 
24,  and  30  respectively. 

15.  A  bell-hanger  wishes  to  find  the  shortest  piece  of  wire 
which  may  be  cut  into  pieces  of  16,  18,  or  22  feet  long. 

16.  What  is  the  least  sum  with  which  a  dealer  can  buy  an 
exact  number  of  hats  at  13,  ^4,  $5,  or  $6  each  ? 

17.  What  is  the  smallest  number  of  gallons  that  can  exactly 
be  measured  by  each  of  4  casks  holding  15,  30,  40,  and  42  gal. 
respectively? 

18.  Two  lads  start  at  the  same  time  and  place  to  travel  round 
a  pond;  one  can  travel  the  distance  in  3  hours,  the  other  in 
4  hours.     In  what  time  will  they  first  meet  at  the  starting-place? 

19.  Three  boats  start  to  sail  round  an  island  at  the  same 
place  and  time ;  one  of  them  can  perform  the  trip  in  6  hours, 
another  in  8  hours,  and  the  other  in  12  hours  ;  how  long 
before  they  will  all  meet  at  the  place  of  starting  ? 

20.  Three  messengers  start  at  the  same  time  from  New 
York  to  go  to  Piiiladelphia  and  back,  one  of  whom  can 
perform  the  journey  in  8  hours,  another  in  10  hours,  and  the 
other  in  12  hours.  In  what  time  will  they  all  meet  at 
New  York  ? 

Questions. 

127.  An  even  number  ?  128.  What  is  an  odd  number  ?  129.  A  prime 
number?     131.  A  composite  number? 

130.  When  are  numbers  prime  to  each  other  ?  132.  What  is  an  exact 
divisor  ?  133.  What  are  factors  ?  134.  A  prime  factor  ?  137.  Name  the 
first  principle  respecting  factors.     Second.     Third. 

138^  Wliat  is  factoring  ?  143.  How  separate  a  number  into  its  prime 
factors  ?  144.  How  find  the  prime  factors  common  to  two  or  more 
numbers  ? 

145.  What  is  a  common  divisor?  146.  The  greatest  common  divisor ? 
149.  How  find  the  g,  c.  d.  ? 

156.  Name  tlie  first  ])rinciple  respecting  multiples.  The  second.  The 
third. 

157.  How  find  the  least  common  inulli]ile  of  two  or  more  numbers  ? 


:^ 


=1^ 


R  A  C  T I O  N  S 


rOUFvTHS. 


Oral     Exercises. 

159.  1.  If  a  unit  is  divided 
into  two  equal  parts,  what  is  eacli 
part  called  ? 

2.  If  divided  into  three  equal 
parts,  what  are  the  parts  called  ? 

3.  If  divided  into  four  equal    ^| 
parts,  what  ? 

4.  When  divided  into  5  equal  parts,  what  are  2  of  the  parts 
called  ?     3  of  the  parts  ? 

5.  When  divided  into   7  equal  parts,  what  is  1  of  the  parts 
called  ?     2  of  the  parts  ?    4  of  the  parts  ?     6  of  the  parts  ? 

6.  If  a  sheet  of  paper  is  divided  into  4  equal  parts,  what 
part  of  the  sheet  is  3  of  the  parts  ?     5  parts  ? 

7.  How  many  halves  in  a  unit  ?     Thirds  ?     Fifths  ? 

8.  Which  is  the  larger,  halves  or  thirds  ? 

160.  A  Fraction  is  one  or  more  of  the  equal  parts  of  a  unit. 

161.  The  Unit  of  a  Fraction  is  the  number  or  thing  of 
which  the  fraction  is  a  part. 

162.  A  Fractional  Unit  is  one  of  the  equal  parts  into  which 
the  number  or  thing  is  divided. 

Thus,  in  the  espressiou  biDO-tliirds  of  a  pear,  the  unit  of  the  fraction  is 
one  pear  ;  and  the  fractional  unit  is  one-third  of  a  pear. 

163.  Fractional  units  take  their  name  from  the  Number  of 
equal  parts  into  which  the  unit  is  divided  ;  as,  thirds,  fourths. 


78  Fraxitions, 

164.  The  numher  of  equal  parts  into  which   the  "unit  Is 
divided  is  called  the  Denominator,  because  it  names  the  parts. 

165.  The  7inm'ber  of  parts  taken  is  called  the  Numerator, 
because  it  numhers  the  parts. 

Thus,  in  the  fraction  three-fourths  (f),  the  denominator  Is  4  and  the 
numerator  is  3. 

166.  The    Terms  of    a    fraction  are    the   numerator  and 

denominator, 

167.  Fractions  are  divided  into  Common  and  Decimal. 

168.  A  Common  Fraction  is  one  in  which  the  unit  is  divided 
into  any  number  of  equal  parts. 

169.  Common  fractions  are  expressed  by  writing  the  de- 
nominator under  the  numerator  with  a  line  between  them. 

Thus,  the  fraction  three-fifths  is  written  f  ;  four-sevenths,  |. 

Express  the  following  fractions  by  figures  : 

Four-tenths. 

Seven-twelfths. 

Two- twentieths. 

Fifteen-thirtieths. 

Twenty-fiftieths. 

Fifty-hundredths. 


83    9  7   _6T_   ,9  9 
10  0?  T2T>  13^0?  ToT* 

110   203   326   5  00 
3  3T?  TBT?  T¥3?  aT4* 

170.  An  Integer  may  be  expressed  in  the  form  of  a  fraction 
by  writing  1  under  it  for  a  denominator. 

Thus,  3  may  be  written  \,  and  read  "  3  ones." 

171.  A  Proper  Fraction  is  one  whose  numerator  is  less  than 
the  denominator  ;  as,  J,  f ,  f. 

172.  An  Improper  Fraction  is  one  whose  numerator  equals 
or  exceeds  the  cienominator ;  as,  f,  f. 


1.  Two-thirds. 

7. 

2.  Three-fourths. 

8. 

3.  Two-fifths. 

9. 

4.  Five-sevenths. 

10. 

5.  Five-eighths. 

11. 

6.  Six-sevenths. 

12. 

Copy  and  read  the  following 

: 

13.     t'i.  iV  T^.  11- 

15 

14.    H,  4il+.'ff. 

16 

Fractions,  79 

173.  A  Simple  Fraction  is  one  whose  terms  are  integers, 
and  may  be  proper  or  improper. 

174.  A  Compound  Fraction  is  a  fraction  of  a  fraction ; 
as,  1  of  }. 

175.  A  Mixed  Number  is  a  ivliole  number  and  a  fraction 
expressed  together  ;  as,  of,  34-|^. 

.176.  Fractions  arise  from  division,  the  numerator  being  the 
dividend,  and  the  denominator  the  divisor.    (Art.  101.)    Hence, 

177.  The  Value  of  a  Fraction  is  the  quotiejit  of  the  numera- 
tor divided  by  the  denominator. 

Thus,  the  value  of  1  fourth  is  l-f-4,  or  | ;  of  6  halves  is  Q-r-2,  or  3  ;  of 
3  thirds  is  3-r-3,  or  1. 

Note. — This  value  depends  upon  the  size  of  the  number  or  thing 
divided,  and  upon  the  number  of  parts  into  which  it  is  divided. 

178.  The  Reciprocal  of  a  Fraction  is  1  divided  by  the 
fraction,  or  the  fraction  inverted.     (Art.  135. ) 

Thus,  the  reciprocal  of  |  is  1  h-  f  =  |  ;  of  y^  is  Y-. 

General     Principles    of     Fractions. 

179.  Since  fractions  arise  from  division,  the  numerator 
being  a  dividend  and  the  denominator  a  divisor,  the  general 
principles  of  division  are  true  of  fractions.  (Art.  119.) 
That  is, 

r.  Mtdtiplying  the  numevatoY,  or  )  T,r  i,-  t     .i     /•      /• 
-r.     . ;.       ^  >  Multiplies  the  fraction. 

Dividing  the  denominator,         ) 

^°.  Dividing    the    numerator,    or)  ^.  .,     ^,    ^ 

■nr  ij-  1   ■      ii      -.  .     i       > Divides  the  fraction. 

Multiplying  the  denominator,  ) 

3°.  Multiplyinq  or  Dividing  both  )  ^  ,      , 

^        "^   /      „      ,.       ,      ,,     (  Does    not    change     its 
terms  oi   a  fraction  by  the  r         , 

,  ''  \      value. 

same  number,  ) 


80  Fractions. 


Reduction    of    Fractions. 

Oral     Exercises. 

180.  1.  How  many  halves  in  a  whole  apple  ?    How  many 

fourths  ? 

2.  How  many  fourths  in  \  of  an  apple  ? 

Analysis. — The  required  denominator  4,  is  twice  the  given  denomina- 
tor 3.     Multiplying  both  terms  of  \  by  2,  it  becomes  |,  An8. 

3.  How  many  sixths  in  f  ?     How  many  ninths  ? 

4.  How  many  eighths  in  f  ?     How  many  twelfths? 

5.  Change  \,  \  to  tenths. 

6.  Change  f,  J,  f ,  -J-,  f  to  sixteenths. 

7.  Change  \,  \,  f,  \,  J,  \,  |  to  twenty-fourths. 

8.  Change  f  to  twenty-eighths  ;  f  to  forty-fifths. 

9.  Change  -^^  to  thirty-ninths  ;  -f-^  to  sixtieths. 

181.  Reduction  of  Fractions  is  changing  their  terms  without 
altering  the  value  of  the  fractions.     (Art.  179,  5°.) 

Written     Exercises. 

182.  To  Reduce  a  Fraction  to  Larger  Terms. 

1.  Change  J4  to  a  fraction  whose  denominator  is  81  ? 

Analysis. — The  given  denominator  27  is  contained  81  -r-  27  =  3 

in  the  required  denominator  81,  3  times.     Multiplying  14x3         42 

both  terms  of  the  given  fraction  by  3,  we  have  |f,  the  -^ -^  =  -^ 

fraction  required.    (Art.  179,  r.)    Hence,  the  ^7  X  ^         bl 

Rule. — Divide  the  required  denojivinator  by  the  denmii- 
inator  of  the  given  fraction,  and  multiply  both  terms 
by  the  quotient. 

2.  Change  ^  to  104ths.  6.  Change  -Jf  to  196ths. 

3.  Change  ||  to  120ths.  7.  Change  f|  to  288ths. 

4.  Change  ^  to  176ths.  8.  Change  -|  to  192ds. 

5.  Change  f|  to  144ths.  9.  Change  ff  to  57Cths. 


Reduction. 


81 


Oral     Exercises. 
183.  Reduce  the  following  at  sight  to  their  lowed  terms : 


1. 
2. 
3. 
4. 


2.     3.      2  _2_  _2_ 

6^    6>    8?  10'  16' 

3         3  3  3  3 

T^>  T'g^?  ¥¥>  "JF?  "&¥• 

_4_      4_  _4        4  4 

12?    f 07  3¥'  18?  6  8* 

■J'B'   "55?  "56?  "63?  "sr* 


An<i     1     1     JL     1     1 

^/*A.     -3,    ^,    ;|,    -g-,    -g. 

5. 


6. 
7. 


-5-  16        8  16     14, 

24?  To?   ^6?  48?    88* 

_9_  13.     lA  IS     _21 

27?  4  5?    6  3?  9  9?    10  8" 

_6_  _6       _6_  _6  6 

18?  24?    4  2?  7^?  "515"* 


184.  A  Fraction   is  expressed  in  its  Lowest  Terms  when 
its  numerator  and  denominator  have  no  common  divisor. 


Written    Exercises. 
185.  To  Reduce  Fractions  to  their  Lowest  Terms. 
1.  Eeduce  -ff  to  an  equivalent  fraction  in  its  lowest  terms. 


1st  Analysis. — Cancelling  the  factor  6  from  both 

terms  of  the  given  fraction,  we  have  3^. 
Again,  cancelling  the  factor  4  from  both  terms  of 
this  fraction,  we  have  f,   the  terms  of  %vhich  are 
prime  to  each  other.    (Art.  130.) 

2d  Analysis. — Dividing  both  terms  of  ff  by 
their  g.  <?.  cl,  24,  (Art.  150)  we  obtain  the  same 
result.     Hence,  the 


1st  operation. 

^  _  A  —  ? 


2d  operation. 

48  -h  24  _  2 
72  -^  24  ""  3 


Rule. — Cancel  all  the  factors  coimnori  to  hotli  terms 
of  the  fraction. 

Or,  Divide  both  terms  hy  their  greatest  common  divisor. 


Change  the  following  to  the  lowest  terms  : 


2. 
3. 
4. 
5. 


6  3 
T6T* 
240 

3  12* 

272 

4  2  5"- 
384 


6. 
7. 
8. 
9. 


eoT* 
3.02 

8"5T* 
25  3 

7  82* 

5  28 
984* 


10. 

11. 

12. 
13. 


750 
^00* 

47_5 
5  2  5"" 
_6JL4_ 
2142* 

5  5 
1210* 


14. 
15. 
16. 
17. 


126 
16  2' 

435 
9"XT* 

1740 

2  9  0  0" 

6465 
7335" 


Oral     Exercises. 

186.     1.  Reduce  -^  to  a  whole  or  mixed  number. 

Analysis. — In  7  sevenths  there  is  1  unit,  and  in  65  sevenths  there  arc 
as  many  units  as  7's  in  65,  or  9f ,  Ans. 


82  Fractions, 

Change  the  following  to  whole  or  mixed  numbers  : 

2.  ^.  5.      ^-.  8.      ^.  11.  -V/. 

3.  V--  6.      -^/.  9.      -3/-.  12.  -\%0-. 

4.  H-  7.       W-.  10.      -8^.  13.  -W-- 

14.     How  many  dollars  in  28  half  dollars  ?    In  36  quarter 
dollars  ?     In  $^  ?     In  %^  ?    In  %^-  ? 

Written    Exercises. 

187.  To  reduce  Improper  Fractions  to  Whole  or  Mixed  Numbers. 
1.  Eeduce  J-^-  to  a  whole  or  mixed  number. 

OPERATION. 

A'NALYSIS. — Since  8  eighths  make  a  unit,  in  1 12  eighths    o  \  1 1  o 

there  are  as  many  units  as  S's  in  112,  or  14  units.    Hence,       <- 

the  14,  Ans, 

Rule. — Divide  the  mnnerator  hy   the  denoniinato7\ 

Note. — If  there  is  a  fraction  in  the  answer,  it  should  be  reduced  to  the 
lowest  terms. 

Reduce  the  following  to  whole  or  mixed  numbers  : 

1  44  8  6  835  ii  _6_18  6  ig  28  3.A2 

2  5  76  7  _7_8_3  lO  8_5'L3  17  98  5  36. 
^-  T3  •  '•  5  5^-  '■^'          4^0-  *■'•  7  50     • 
•a  JL5  0  Q  84  3  7  1 Q  _9  5  6  8  i  o  .10  0  0  0  0 
«•  -2^-  °'  "298  •  ^^-          2T^-  ^°-  59000   • 
A  -8JL5  Q  1_2_4_3  14  1.2  0.0JO  iq  A_1^0J)31 
4.  3Q  .  V.          32  0'  ^^-            12  1     •  ^^-  7  2  1T6- 

K  SOO  10  5-8-0-S  IK  15  720  OQ  8JL9^2JI 

°*  16"*  ^^'  126'  •^°*  1T68*  ^^'  72840* 

21.  In  ^||g^  of  a  rod  how  many  rods  ? 

22.  In  -f  gfJ^  ^f  '^  dollar  how  many  dollars  ? 

Oral     Exercises. 

188.  1.  Reduce  7f  to  an  improper  fraction. 

Analysis, — Since  in  1  unit  there  are  4  fourths,  in  7  units  there  are 
7  times  4,  or  28  fourths,  and  3  added  make  31  fourths.     Ans.  \^-. 

Reduce  the  following  to  improper  fractions  : 

2. 
3. 
4. 

14.  Change  7  to  nintlis  ;  8  to  sevenths  ;  11  to  eighths. 

15.  Change  14  to  thirds  ;  12  to  ninths  ;  15  to  fourths. 


4i. 

5. 

^i- 

8. 

llf. 

11.   15f. 

5}. 

6. 

8|. 

9. 

^. 

12.  20f 

n- 

7. 

9f. 

10. 

12f 

13.  25|. 

Common  Denominators,  83 

Written     Exercises. 
189.  To  reduce  Whole  or  Mixed  Numbers  to  Improper  Fractions^ 

1.  Reduce  18 J  to  an  improper  fraction. 

Analysis. — Since  in  1  there  are  5  fifths,  in  18  there  are      ^"T 
18  times  5  fifths,  or  90,  and  3  fifths  added  make  93  fifths.         5 
Am.  ¥•    Hence,  the  "  g^^j^^^ 

Rule. — Multiply  the  whole  niniiber  hy  the  given  clenom-- 
inator ;  to  the  product  add  tlie  lunnerator,  and  place, 
the  sum  over  the  denominator. 

Note. — 1.  A  whole  number  is  reduced  to  an  improper  fraction  by  making; 
1  its  denominator.     Thus,  4  =  f .     (Art.  170.) 

2.  For  reducing  a  Compound  Fraction  to  a  Simple  one,  See  Art.  211. 

Reduce  the  following  to  improper  fractions  : 

2.  Reduce  19f.  8.  Reduce  26|. 

3.  Reduce  23|. 

4.  Reduce  64^. 

5.  Reduce  304^. 

6.  Reduce  45  to  fifths. 

7.  Reduce  830  to  sixths. 


Common    Denominators. 

Oral     Exercises. 

190.  1.  Change  -J-  and  f  to  fractions  whose  denominator  is  6.. 

Analysis. — Multiplying  both  terms  of  |  by  8,  we  have  f  ;  and  multiply- 
ing both  terms  of  f  by  3,  we  have  |.     Ans.  f  and  f . 

2.  By  what  must  ^  and  f  be  multiplied  to  become  twelfths  ?.' 

3.  Change  -|  and  |-  to  the  same  denominator. 

4.  Change  }  and  ^  to  the  same  denomi;»iator. 

5.  Name  two  multiples  of  3  ;  of  4  ;  of  6  ;  of  7. 

6.  Name  a  multiple  common  to  3  and  5. 

7.  Change  \  and  f  to  twenty-fourths. 

191.  Fractions  which  have  the  same  denominator,  hare  a 
Common  Denominator, 


9. 

Reduce  45^. 

10. 

Reduce  56}f. 

11. 

Reduce  725|. 

12. 

Reduce  72  to  eighths. 

13. 

Reduce  743  to  15ths. 

84  Fractions. 

192.  The  Least  Common  Denominator  (?.  c.  ^.)  of  two  or 

more  fractions^  is  the  smallest  number  divisible  by  each  of 
their  denominators. 

193.  The  smallest  number  divisible  by  any  two  or  more 
numbers  is  their  Least  Common  Multiple.    (Art.  153.)    Hence, 

194.  The  Least  Common  Denominator  of  two  or  more  frac- 
tions, is  the  Least  Common  Multiple  of  their  denominators. 

Note. — When  the  denominators  are  prime  to  each  other,  their  co";* 
tinued  product  is  their  I,  c.  d. 

Written     Exercises. 

195.  To  Reduce  Fractions  to  a  Common  Denominator. 

1.  Eeduce  ^,  |,  and  f  to  equivalent  fractions  having  a  c,  d. 
Analysis.— The  product  of  the  de-         .J,  |.,  ^^  Given  Fraction, 
nominators  2,  3,  and  5,  is  30,  the  com-  2  X  3   X  5  =  30    C,  cl* 

mon  denominator.     (Art.  191.)  1   X  3   X  5  ==  is'  1st.  A. 


2  X  2  X  5  =  20,  2d  n. 


Multiplying  each  numerator  by  all 
the  denominators  except  its  own,  we 
have  if,  If,  14.    (Art.  179,  3\)    Hence,  4  X  2  X  3  =  24,  3d    il, 

tne  j^nc.    3Q,   -g-Q,    3Q. 

Rule. — Multiply  the  clenominators  together  for  the 
cojmnoji  denominator,  and  each  numerator  by  all  the 
denominators,  except  its  own,  for  the  numerators. 

Reduce  to  a  common  denominator, 

2.     f  and  f .  4.     I  and  ^.  6.     ||,  J|,  and  tItt- 

196.  To  Reduce  Fractions  to  the  Least  Common  Denominator. 

8.  Reduce  f ,  f,  and  -^,  to  their  I.  e,  d. 

Analysis.— Reducing  the  given  3  X  3  X  5  =  45,  7.  C.  cZ. 

denominators  to  the  I,  c.  Ji/.,  we  45_:_3  ^^  15   ^nd   2.X15  — -  ao 

have  45  for  the  I.  c,  d,     (Art.  157.)  t-   .  n           k   ^    a      r^k         9  k 

Multiply  the   terms  of  each   frac-  ^^        *^ 

tion  by  the  quotient  of  45  divided  45-^15  =     3   and  -l^^g  =z  |^ 

by  each  given  denominator,   and  the  products  will  be  ff ,  ff ,  |f,  Ans. 
Hence,  the 

RcTLE. — Find  the  least  common  multiple  of  all  the 
denominators  for  the  least  common  denominator. 


Addition,  85 

Divide  this  muUiple  hy  the  denominator  of  each  frac- 
tion, and  -multiply  its  numerator  hy  the  quotient. 

Note. — Mixed  numbers  must  be  reduced  to  improper  fractions  and  all 
to  their  lowest  terms  before  the  rule  is  applied. 

Reduce  the  following  fractions  to  the  I,  c.  d, : 

9.      I,   tV.    h'  15.      A,    1%   If. 

10.  I,  I,  A-  16.     ^,  8f,  ■^. 

11.  tV.   A-.  tV  17.     ^,  I,  2f. 

12.    h  h  tV  a.  18.    11.  lOi  H- 

13.  tV^   -f.   H.   f-  19-      ^A.   ih   1%' 

14.  if,  ft,  H-  20.      ft,    ^V    AV 


Addition    of    Fractions. 

Definitions. 

197.  Like  Fractions  are  those  which  express  likti  parts  of 
like  units. 

Thus,  I  yard  and  |  yard  ;  also  f  and  |  are  like  fractions. 

198.  Unlike  Fractions  are  those  which  express  unlike  parts 
of  like  units,  or  parts  of  unlike  units.     (Art.  7.) 

Thus,  I  pound  and  f  pound  ;  also  4  and  f  are  unlike  fractions. 

Oral     Exercises. 

199.  1.  What  is  the  sum  of  f ,  \,  and  f  ? 
Solution. — f  and  \  are  f ,  and  f  are  f ,  Ans. 

2.  What  is  the  sum  of  ^  +^  +  ^  ?    Of  A  + A+ A  ? 

3.  A  dealer  sold  -{-^  ton  of  coal  to  one  person,  -^  to  another, 
and  -^  to  another  ;  how  much  coal  did  he  sell  to  all? 

4.  A  Reader  costs  %^  and  a  History  $f ;  what  will  both  cost? 

Analysis. — Halves  and  fourths  are  unlike  parts  of  the  unit  dollar,  and 
cannot  be  added  in  their  present  form.  But  ^  is  equal  to  f,  and  f +  f  are 
I,  which  equals  $1-^,  Ans.     (Art.  197.) 

5.  What  is  the  sum  of  f  and  -|-  ?     Of  f  and  |  ? 

6.  How  much  wood  is  there  in  |  cord  and  \-  cord  ? 


86  Fractions. 

7.  What  is  the  sum  of  -|  pear  and  \  melon  ? 
Ans.  Pears  and  melons  are  unlihe  units,  and  parts  of  unlike 
units  cannot  be  added.     (Art.  53,  i°.)     Hence,  the  following 

Principles. 

200.  1°.  Only  like  fractions  can  he  added.     (Art.  197.) 
2°.  Like  fractions  are  added  the  same  as  like  integers. 

8.  I  and  ^=  ?         11.     I  and  -f^=  ?         14.     ^^  and  i^=  ? 

9.  J-  and  1=  ?        12.     f  and  -f-^=  ?        15.     f  and  i^=:  ? 
:io.     fand|=?        13.     |andii=?        16.    35_andTS-^? 

:  17.  A  farmer  sold  -f  tons  of  hay  to  one  neighbor,  and  ^  of 
"a'ton  to  another ;  how  much  hay  did  he  sell  to  both? 

18.  If  a  newsboy  makes  IJ  in  one  day,  ||-  the  next,  and  If  the 
^hird  day,  how  much  will  he  make  in  3  days  ? 

19.  If  a  pupil  is  absent  -J  day  1  week,  -|  day  the  next,  and  f 
day  a  third  week,  how  much  time  has  he  lost  in  3  weeks. 

Written     Exercises. 

201.  To  find  the  Sum  of  two  or  more  Fractions. 

1.  What  is  the  sum  of  |,  f,  and  ^  ? 

Analysis. — Reducing  the  given  fractions  "S  ^^  "^ 

to  the  I.  c.  d.  24,  they  become  -i^.  If,  and  I  =  If 

\\,  which  are  like  fractions,  the  sum  of  whose  -^^  =  -|^ 

numerators  is  43  ;    and  41  =  141,  the  answer  9^  _|_  2_q.  _i_  i  jl  —  A3 

required.     Hence,  the  43  -m,      j„- 

EuLE.  —  Reduce  the  given  fractions  to  a  common 
denominator,  and  over  it  write  the  sum  of  their  nu- 
merators. 

Notes. — 1.  If  there  are  mixed  numbers,  add  the  fractions  and  integers 
separately,  and  unite  the  results. 

2.  The  answer  should  be  reduced  to  lowest  terms,  and  if  improper 
fractions,  to  whole  or  mixed  numbers. 

3.  It  is  advisable  in  most  cases  to  reduce  the  fractions  to  the  L  c,  d. 


Add  the  following  : 

3. 

1,  -^,  and  i«5. 

4. 

-/^,  1,  and  if. 

5. 

i,  f,  f,  and  -f. 

6. 

-rV,  f ,  and  f 

7. 

1,  i    |,  and  |. 

8. 

1,  f,  i,  and  T|L. 

9. 

i  i  h  i  and  J. 

10. 

2J-,  6f,  and  f. 

11. 

If,  11,  and  \l 

Addition,  87 

2.  What  is  ihd  sum  of  12|,  19|,  and  15  ? 

Analysis. — Reducing    the    fractional    parts  to  a  "i"  ^^  -'^'^iV 

common  denominator,  §  and  f  are  equal  to  y*,  and  -^.  19|   =  l^iT 

Now  y^2  +  y^g  =  \l,  or  1/3.     Adding  the  1  to  the  sum  15     =15 
of  the  integral  parts,  we  have  47  4-  ,%-  =  47 A,  ^ws.  ^  .^  - 


12.  1,  3,  I,  i-,  and  |. 

13.  f ,  2,  31   and  5f . 

14.  351  fl,  1^  and|. 

15.  ^,  ^,  If,  and  |. 

16.  i\,  85,  fi,  and  3f . 

17.  24|,  82f,  and  if. 

18.  25|,  181^,  and  |f. 

19  3.3J.      3J.5.     or,r]    5  88 

'■^'        448^    693?    «-*^^^  T56' 

20.     263|,  f,  f I,  and  385f. 

21.  A  grocer  sold  47f  pounds  of  sugar  to  one  customer,  83| 
pounds  to  another,  and  G8f  pounds  to  another ;  how  much  did 
he  sell  to  all  ? 

22.  If  you  travel  85  j^g-  miles  in  one  day,  '^"^8^  in  another,  and 
125JI  in  another,  how  far  will  you  travel  in  all  ? 

23.  If  a  man  buys  3  pieces  of  cloth,  containing  127-|  yards, 
1^8^  yards,  and  256|-  yards,  how  much  will  he  then  have  ? 

24.  If  a  hat  costs  $4f  and  a  vest  $5|-,  what  will  both  cost  ? 

Analysis. — $4  and  $5  are  $9 ;  i  =  f ,  and  f  +  |  are  f  —  $1^,  which 
added  to  $9  make  $10|,  Ans. 

of  6|  and  5^? 

30.  15i|-  and  21}  ? 

31.  7x5j-  and  15^'V? 

32.  20yV  and  ^j%  ? 

33.  30f  and  20f  ? 

34.  A  man  bought  3  pieces  of  cloth,  one  of  which  contained 
45f  yards,  another  63|  yards,  and  the  other  564-  yards  ;  bow 
many  yards  did  he  buy  ? 

35.  If  you  travel  85|  miles  in  one  day,  95^  miles  the  second, 
and  115/g-  miles  the  third  day,  how  far  will  you  travel  in  all? 


25. 

What  IS  the  s 

26. 

91-  and  7|  ? 

27. 

15|  and  18J  ? 

28. 

71  and  9| ? 

29. 

12f  and  20f ? 

88  Frcbctions, 

Subtraction   of    Fractions. 

Mental    Exercises- 

202.  1.  What  is  the  difference  between  f  and  I  ?    (Art.  197.) 

2.  Find  the  difference  between  ^^  and  ^^.     \\  and  -f-^. 

3.  If  you  have  -J  pound  of  caudy  and  give  away  f,  how  much 
will  you  have  left  ? 

4.  George  had  l^^  and  gave  I/q-  for  a  Eeader,  how  much 
money  did  he  have  left  ? 

5.  What  is  the  difference  between  %\  and  %\  ? 

Analysis. — Thirds  and  sixths  are  unlike  parts  of  the  unit  $1,  and  one 
cannot  be  taken  from  the  other  in  their  present  form.  But  \  is  equal  to  |, 
and  f  from  |  leave  f ,  or  %\,  Ans. 

6.  Florence  has  two  pieces  of  ribbon,  one  is  f  of  a  yard  long 
and  the  other  |  of  a  yard  ;  what  is  the  difference  in  length  ? 

7.  If  a  cap  costs  %\  and  a  pair  of  mittens  If,  what  is  the 
difference  in  their  price  ? 

8.  Wliat  is  the  difference  between  |  quart  and  |  yard  ? 

Analysis. — Quarts  and  yards  are  unlike  units,  and  parts  of  one  can- 
not be  taken  from  parts  of  the  othero 

203.  From  the  above  examples  we  infer  the  following 
Prii^ciple. —  Onlij  like  fractions  can  he  subtracted. 

9.  What  is  the  difference  between  |  and  |  ? 

10.  What  is  the  difference  between  f  and  |  ? 

11.  One  man  owned  f  of  a  ship  and  another  f ;  what  wa^ 
the  difference  in  their  ownership  ? 

12.  Richard's  kite-line  was  18|  yards  long  and  he  cut  off 
Ct¥  y^i'ds ;  how  long  was  the  part  left  ? 

Analysis. — f^  are  equal  to  |,  and  \  from  |  leaves  f .  Again,  6  yd.  from 
18  yd.  leave  12  yd.,  and  |  added  make  13|  yards,  Ans. 

Required  the  difference  between 

13.  f  and  |.  17.  -^  and  f .  21.  8}  a-nd  5J. 

14.  I  and  f.  18.  -jAj-  and  if.  22.  15|  and  18|. 

15.  ^  and  |.  19.  y'Jj  and  ||.  23.  25|  and  20f . 

16.  J  and  f .  20.  ^  and  |^f .  24.  344  and  25^. 


Subtraction,  89 

Written     Exercises. 
204.  To  find  the  Difference  between  two  fractions. 

1.  From  I  subtract  f. 

Analysis. — Reducing  the  given  fractions  to  operation. 

the  I,  c,  d,  18,  they  become  J  |  and  -f^,  and  |-  =  -l-|  ;  |  =  ^ ; 

H— A  =  A.  Ans.  H— T%  =  tV  ^'^^' 

2.  From  246|  subtract  132J. 

Analysis.— Reducing  the  fractions  |  and  f  to  the  operation. 

I.  c,  d.  20,  we  have  f  =  /o>  f  =  M-     Now  to  sub-  246|  =  246^ 

tract  U  from  ^»o,  we  take  1  =  (|g)  from  6,  and  add  it  132|  —  .132||- 

to  ^^^,  making  |f,  then  subtract,  and  take  2  from  5,  TTqTT 

etc.     Hence,  the  ^^^^'   ^^^^ 

Rule. — /.  Reduce  the  given  fractions  to  a  com-mon 
denominator,  and  ovei'  it  write  the  difference  of  the 
numerators.     (Art.  195.) 

II,  If  there  are  mixed  numbers,  subtract  the  frac- 
tional and  integral  parts  separately,  and  ivniUy  the 
results.     (Ex.  2.) 

Note. — In  most  cases  it  is  better  to  reduce  the  fractions  to  the  least 
common  denominator. 

Wliat  is  the  difference  between 


3. 

H  and  i^A- 

8. 

M  and  ^. 

13. 

2  and  |. 

4. 

■h  and  ,\. 

9. 

2  and  ^2^. 

14. 

65  and  25f . 

5. 

2  1    qnrl    2  3 

3T  ana  4  8 . 

10. 

+4  and  if. 

15. 

21f  and  9f 

6. 

S^  and  of. 

11. 

12|  and  8|. 

16. 

25|  and  17f. 

7. 

121-  and  7^. 

12. 

15f  and  9J. 

17. 

37f  aud  19i. 

4   ? 


18.  From  385J  rods  take  67J  rods. 

19.  From  573|  tons  take  21 6|  tons. 

20.  From  563J  pounds  take  260|^  pounds. 

21.  From  1673-1  bushels  take  356f  bushels. 

22.  A  man  bought  a  wagon  for  -$851,  and  a  sleigh  for  $69 
how  much  more  did  he  pay  for  one  than  the  other  ? 

23.  A  man  having  246i^g  acres  of  land,   sold  195f  acres  ; 
how  many  acres  did  he  have  left  ? 

24.  If  from  a  piece  of  cloth  containing  125|-§^  yards,  you  cut 
87^  yards,  how  many  yards  will  be  left  ? 


90  JFraotions, 

Multiplication    of    Fractions. 

Oral     Exercises. 

205.  1.  What  is  the  cost  of  5  books,  at  $|  apiece  ? 

ANA1.TSIS. — Since  1  book  costs  $|,  5  books  will  cost  5  times  $f,  which 

are  $V-  =  $1|»  ^''^^^ 

2.  What  cost  6  bushels  of  apples,  at  $y\-  a  bushel  ? 

3.  At  %-^^  a  pound,  what  will  10  pounds  of  butter  come  to  ? 

4.  Multiply  f  by  8.  5.  Multiply  J  by  12. 

6.  How  many  units  in  7  times  4-  ? 

7.  At  1^  a  pound,  what  will  4  pounds  of  tea  come  to  ? 

Analysis. — Dividing  the  denominator  12  by  4,  multiplies  the  fraction 
(Art.   179),  the  result  is  $V  =  $3f ,  Ans. 

8.  Multiply  I  by  3.  li.  Multiply  /-j  by  7. 

9.  "        -^  by  5.  12.  "        -^  by  10. 
10.          «        iVl^y^.                 13,  «        if  by  15. 

14.  AYhat  will  4  yds.  of  braid  come  to,  at  5|  cts.  a  yard? 

Analysis. — Since  1  yd.  is  worth  5|  cents,  4  yards  are  worth  4  times 
5|  cents.  Now  4  times  5  are  20  cts.  and  4  times  2  thirds  are  8  thirds 
equal  to  2|  cents,  which  added  to  20  make  22f  cents.     Therefore,  etc, 

15.  At  6 J  cents  each,  what  must  I  pay  for  8  oranges  ? 

16.  At  $5f  a  yard,  what  is  the  cost  of  7  yds.  of  cloth  ? 

17.  What  must  a  lady  pay  for  8  yds.  of  silk  at  |3|  a  yard  ? 

18.  How  many  are  7  times  8|-? 

19.  Multiply  lOf  by  8. 

20.  What  is  the  product  of  12|  by  9  ? 

Written     Exercises. 

206.  Multiplying  a  Fraction  by  an  Integer. 

A  Fractio7i  is  multijMecl  hy  nniltiplyinci  Us  numerator  or 
hy  dividing  its  denominator.     (Art.  179,  1°.) 


Multiplication.  91 

1.  Multiply  J^  by  9. 

Explanation.  —  Multiplying  1st  opekation. 

the  numerator  13  by  9,  the  result  ^|  ^  9  r=  -iji/  —  2|,    Ans, 

equals  3|.  Or,   M  X  ^  =   ¥   =  -i 

Cancelling  the  factor  9,  which                     g 
is   common   to   both  terms,    the                               2d  operation. 
result  is  the  same.  j^         jLJ.   9|.     Ans» 

Or,  dividing  the  denominator  45-^9  5  1' 

45  b,y  9,  the  result  is  3|,  as  before. 

Note. — In  the  1st  operation,  the  number  of  parts  is  increased,  while 
their  size  is  unchanged.  In  the  2d  operation,  the  size  of  the  parts  is 
increased,  while  their  number  is  unchanged. 

2.  Multiply  ^   by  14.  ^^5.  3|. 

3.  ^x9  =  ?         6.    Ax45==?  9.    1^0^x10=:? 

4.  ^Vxl2z=?       7.     AVx48=?       10.     f/^x41=? 

5.  ifX^l  =  ?         8.     -r*^x86==?         11.      22_.5.^  X  54  =:  ? 

12.     Multiply  15f  by  7. 

152- 
ExPLANATiON.  —  Multiply    the  fractional    and  integral  * 

parts  of  15|  separately,   and  uniting  the  results,  we  have  ' 

110},  the  product  required.  Ans.   110 J 

Note. — When  the  multiplicand  is  a  mixed  number,  the  fractional  and 
integral  part  should  be  multiplied  separately,  and  the  results  be  united. 

13.  87ix8=?  15.  205|x24=:?  17.  256f  x3  =  ? 

14.  165|xl2=?      16.  196yVxl8  =  ?        18.  575yVx48=r? 

Oral    Exercis  es. 
207.  When  the  Multiplier  is  a  Fraction. 

1.  If  a  barrel  of  flour  is  worth  $6,  what  is  J  barrel  worth  ? 

Analysis. — 1  half  barrel  is  worth  1  half  as  much  as  a  whole  barrel, 
and  1  half  of  $6  is  $3.     Therefore,  etc. 

2.  If  a  stage  goes  9  miles  an  hour,  how  far  will  it  go  in  \  of 
an  hour? 

3.  What  is  A  of  14  apples  ?    1  of  15  pounds  ?    ^  of  28  days  ? 

4.  At  $5  a  yard,  what  Avill  J  of  a  yard  of  cloth  cost  ? 

Analysis. — |  of  a  yard  will  cost  f  times  $5,  or  3  times  \  of  $5.  Now 
\  of  $5  —  $1^,  and  3  times  $1^  are  %^. 


92  fractions, 

5.  Whatis  Jof  I? 

Analysis. — |  of  f  are  equal  to  3  times  \  of  |.    Now  |^  of  f  is  /^,  and 
3  fourtlis  are  3  times  /g  ^^  if  =  T2>  ^^*- 

6.  At  If  a  pound,  what  will  f  pound  of  tea  cost  ? 

7.  What  costs  "I  of  a  box  of  lemons,  at  $G  a  box  ? 

8.  At  $8  a  barrel,  what  will  |  of  a  barrel  of  flour  cost  ? 


9. 

fof  12=? 

12. 

fof  42feet  =  ? 

15. 

|of  60=? 

10. 

-f  of  13=? 

13. 

1  of  40  yds.  =  ? 

16. 

T^of  72  =  ? 

11. 

1  of  16  =  ? 

14. 

y6TOf25lbs.  =  ? 

17. 

/o  of  200  =  : 

18.  At  8  cts.  a  yd.  what  will  be  the  cost  of  5f  yds.  of  muslin ; 

Analysis. — 5|  yds.  will  cost  5|  times  8  cts.  Now  5  times  8  cts.  are 
40  cts. ;  1  fourth  of  8  cts.  is  3  cts.  and  3  fourths  are  3  times  2  cts.  or  6  cts. 
which  added  to  40  cts.  make  46  cents.     Therefore,  etc. 

19.  At  7  cts.  a  pound,  what  will  5 J  pounds  of  sal  soda  cost  ? 

20.  How  many  are  8f  times  9  ? 

21.  How  many  are  7f  times  12  ? 

22.  At  6  shillings  a  pound,  what  cost  5f  pounds  of  tea. 

23.  What  cost  7f  acres  of  land,  at  $10  per  acre  ? 

Written     Exercises. 

208.  Multiply i7ig  hy  a  fraction  is  tahing  a  certain  part 
of  the  tmdtiplicand  as  many  times,  as  there  are  like  parts  of  a 
unit  in  the  multiplier.    Thus, 

Multiplying  by  ^,  is  taking  1  half  of  the  multiplicand  once. 

Multiplying  by  ^,  is  taking  1  third  of  the  multiplicand  once. 

Multiplying  by  f ,  is  taking  1  third  of  the  multiplicand  tivicc. 

Note. — 1.  To  find  a  halfoi  a  number,  divide  it  hy  2.  To  find  a  if!iird  of  a 
number,  divide  it  hy  3.     To  find  ix  fourth  of  a  number,  divide  it  hy  4,  etc. 

1.  Multiply  63  by  f 

Analysis.— Multiplying  63  by  |,  is  find-  ^-  —  ^^  =  36,  Ans. 
ing  I  of  63.    Now  -|  of  03  =  4  times  ^  of  63  ;  9 

I  of  63  is  9  and  j  is  4  times  9  =  36.  '  Or,     ^^-^  =  36,   A71S. 

Or,  cancelling  tire  factor  7,  common  to  both  terms,  we  have  9  x  4  =  36. 


Midti'plication.  93 


Note. — 2.  A  fraction  is  multiplied  by  a  number  equal  to  its  denomi- 
nator \>j  cancelling  its  denominator.    (Arts.  123,  r  ;  179,  1°.) 

In  like  manner  a  fraction  is  multiplied  by  any  factor  of  its  denomi- 
nator by  cancelling  that  factor. 

Find  the  product  of 

2.  60  by  /_  5.     112  by  ^^,  8.     39  by  if. 

3.  63  by  ^§j..  6.     168  by  l^.  9.     896  by  |^. 

4.  70  by  T^.  7.     105  by  V-  10-     572  by  J-Q. 

11.  Multiply  160  by  5|.    * 

OPERATION. 

Note.  —  3.    When  the    multiplier   is    a  mixed         IgQ  x  —  =  120 

number,    multiply  by  the   fractional  and  integral  i  p^        p-  o^rj 

parts  separately  and  unite  the  results.  ^  

^?i5.  920 

Multiply  the  following  : 

12.  93  by  12f .  16.  256  by  17^-  20.  107  by  47|J. 

13.  184byl8|.  17.  196by41ii.  21.  510  by  85i|. 

14.  125  by  10^6^.  18.  341  by  30yV  22.  834  by  89^1^. 

15.  268  by  12Jf .  19.  457  by  12ff  23.  963  by  951^. 

Oral     Exercises. 

209.  1.  If  I  cut  J  sheet  of  paper  into  2  equal  parts,  what 
part  of  1  sheet  will  there  be  in  each  piece  ? 

Ans.  ^  of  a  /m//* sheet,  which  is  equal  to  J  sheet. 

2.  If  a  bushel  of  apples  costs  %^,  what  will  J  bushel  cost? 

Analysis. — i^  bushel  will  cost  \  as  much  as  a  whole  bushel ;  and  \  of 
$i  is  $i.     (Art.  179,  2° .) 


3.  What  part  of  1  is  |  of  |  ? 

4ofi?    iofi? 

4.  Whicli  is  greater  J  or  ^  ? 

JrOrJj?    |or-,V? 

5.  What  is  1  off?   |of  1  ? 

|of-ft?    4  of  A? 

6.  If  a  pound  of  tea  is  worth  $|,  what  is  f  pound  wortk  ? 

7.  What  cost  -J  yard  of  ribbon,  at  ^f  a  yard  ? 

8.  If  a  yard  of  cashmere  is  worth  -^f ,  what  is  f  yd.  worth  ? 

9.  A  man  owning  f  of  a  yacht,  sold  J  of  it  to  his  neighbor  ; 
what  part  of  the  yacht  did  each  then  own  ? 


94  Fractions, 

Written     Exercises. 
210.  Multiplying  a  Fraction  by  a  Fraction. 

1.  At  $f  a  yard,  what  will  J  yd.  of  silk  cost  ? 

Explanation.  —  One-fourtli   of  a  yard  operation. 

will  cost  \  as  much  as   1  yard,  and  \  ot  $-|  X  |  =  f^  or  If. 


$/g,  and   3  fourths  yard  will    cost  3  ^ 


times  %i^,  or  $ff  =  ||,  ^7^«.  Or,   |  X  f  =r  $f ,  Ans. 

Or,  indicating  the  operation,  and  cancel-  3 

ling  the  factors  common  to  the  terms  of  the  fractions,  we  have  $|  the 
same  as  before. 

Note. — The  above  solution  is  the  same  in  effect  as  multiplying  the 
numerators  together  for  the  numerator,  and  the  denominators  for  the 
denominator  of  the  required  product. 

2.  Multiply  i  of  I  of  I  by  i  of  ^. 

Explanation. — The  product  of  the  numerators  f ,  |,  f ,  ^,  y^,  is  120 ; 
the  product  of  the  denominators  is  1440  ;  and  y  4V0  =  tV»  Ans. 

Or,  cancelling  the  factors  common  to  the      |.xfxl-Xi^XT\  ^^ 

numerators  and  denominators,  the  result  is      Xs/^v/'5.viv_4    _i_ 

3*2,  the  Ans.  required. 

211.  The  word  of,  iu  Compound  Fractions,  has  the  force  of 
the  sign  of  multiplication  x .  Multiplying  compound  fractions 
together  reduces  them  to  2i_  simple  fraction. 

Thus,  f  of  f  of  I  is  a  compound  fraction,  and  is  equivalent  to  f  x  |  x  |, 
which  is  equal  to  l^,  a  simple  fraction. 

3.  Multiply  I  of  4  by  |  of  -|  of  14.  Ans.  f|  =  2||. 

4.  At  |6|-  a  barrel,  what  are  5^  barrels  of  flour  worth  ? 

Note. — If  either  factor  is  a  mixed  operation. 

or  whole  number  it  may  be  reduced  6|-  =  -^  and  5-^  =  ^^ 

to  an   improper  fraction,   and    the  5  1  v^   1 6_  _-  432  __  ^3^     AnS 

operation  becomes  the  same  as  mul-  9            4 

tiplying  a  fraction  by  a  fraction.  Or,      ^^  X  -^^  ==  $36,   Ans, 

5.       -Axi}  =  ?  9.       fof|offo=:? 

6.  «xH  =  ?  10.     tfofifofH^? 

7-  t\xH  =  ?  11.      f  of  25xf  of  J=z:? 

8.  |x|x|  =  ?  12.     fof  30x-Hof  |  =  ? 

i3.  IIow  many  ^re  j^  of  45  x  |^  of  |  ? 


Multiplication,  95 

14.  If  a  quart  of  chestnuts  costs  |^  of  j  of  40  cents  what  will 
-J  of  ^  of  a  quart  cost  ? 

15.  What  cost  15-|  tons  of  coal,  at  $6f  a  ton  ? 

212.  The  preceding  principles  may  be  summed  up  in  the 

following 

General     Rule. 

Reduce  whole  and  mixed  ninnhers  to  improper  frac- 
tions, compound  fractions  to  simple  ones,  and  cancelling 
the  common  factors,  lurite  the  product  of  the  numerators 
over  the  product  of  the  denominojtors. 

A-PPLICATIONS. 

213.  1.  At  l-l  a  cord,  how  much  will  the  sawing  of  20J  cords 
of  wood  amount  to  ? 

2.  What  cost  1 6  pounds  of  cheese,  at  8^  cents  a  pound  ? 

3.  What  cost  9  dozen  of  eggs,  at  12-|-  cents  per  dozen  ? 

4.  What  cost  lof  yards  of  cambric,  at  15  cents  per  yard? 

5.  What  cost  111  cords  of  wood,  at  |3|-  per  cord  ? 

6.  At  12 J  cents  a  pound,  what  cost  2f  pounds  of  pepper? 

7.  What  cost  18  ounces  of  nutmegs,  at  16 J-  cts.  an  ounce  ? 

8.  At  12f  cents  a  yard,  what  will  27  yards  of  cotton  cost? 

9.  At  $3^|-  a  yard,  what  cost  15^  yards  of  broadcloth  ? 

10.  What  cost  15}  yards  of  ribbon,  at  40  cents  per  yard  ? 

11.  What  cost  22  penknives,  at  |^|  apiece? 

12.  At  %^-Q  a  yard,  what  cost  8}  yards  of  silk  ? 

13.  At  $1  a  yard,  what  will  9^  yards  of  muslin  cost  ? 

14.  At  If  a  bushels  what  cost  TyV  bushels  of  wheat? 

15.  What  will  8-f  pounds  of  tea  cost,  at  $f  a  pound  ? 

16.  What  cost  66  bushels  of  apples,  at  18|  cents  a  bushel  ? 

17.  At  32^  cents  a  yard,  what  cost  12|  yards  of  gingham  ? 

18.  What  cost  18-|  yards  of  lace,  at  16^^  cents  per  yard  ? 

19.  What  cost  43  bushels  of  oats,  at  18}  cents  a  bushel? 

20.  What  cost  31J  yards  of  sheeting,  at  $f  per  yard? 

21.  At  $y\  a  quart,  what  cost  18-|-  quarts  of  cherries? 

22.  What  cost  14|  bushels  of  potatoes,  at  18}  cents  a  bushel? 

23.  At  $-|  a  yard,  what  cost  8|  yards  of  velvet  ? 

24.  At  8-J  a  bushel,  what  costs  47|-  bushels  of  pears  ? 


96  Fractions. 

25.  What  cost  63f  pounds  of  sugar,  at  9f  cents  per  pound  ? 

26.  What  cost  22|-  yards  of  velvet,  at  |3f  a  yard  ? 

27.  What  cost  25^  pounds  of  figs,  at  Ib^  cents  a  pound  ? 

28.  What  cost  35|  cords  of  wood,  at  $3|  per  cord  ? 

29.  What  cost  175 J  bushels  of  corn,  at  If  a  bushel  ? 

30.  What  cost  38|  tons  of  hay,  at  115-J  a  ton  ? 

31.  At  42|^  miles  a  day,  hoAV  far  can  you  travel  in  17^  days  ■ 

32.  Mult.  126  by  |  of  33.  37.  Mult.  -||f  by  ^  of  |f|. 

33.  Mult,  f  of  9  by  f  of  7.  38.  Mult,  ff  by  l^. 

34.  Mult,  f  of  184-  by  f  of  241-.  39.  Mult.  -|  of  |  by  f  of  f. 

35.  Mult.  217i  by  |  of  f  of  8.  40.  Mult.  16f  by  f  of  6. 

36.  Mult.  Ill  by^l  of  HI-  41.  Mult.  468t5j.  by  j  of  f^-o 

42.  Multiply  f  of  I  of  -j^  of  if  of  11  by  1  of  f  of  45. 

43.  Multiply  I  of  A  of  If  of  i  of  29  by  H  of  y^o  of  A- 

44.  Multiply  I  of  If  of  -^  of  16i  by  ||  of  ||  of  |  of  49, 


Division    of    Fractions. 

Oral     Exercises. 

214.     1.  If  2  melons  cost  If,  what  will  1  melon  cost  ? 

Analysis. — 1  melon  is  |  of  3  melons  ;  therefore,  1  melon  will  cost  ^  of 

$1,  and  I  of  $4  is  $f,  Ans. 

2.  If  3  knives  are  worth  %-^-q,  what  is  1  knife  worth  ? 

3.  K I  pay  1^  for  4  slates,  what  do  I  pay  for  1  slate  ? 

4.  If  2  pears  cost  -|  of  a  dime,  how  much  will  1  pear  cost? 

5.  If  J  of  a  yard  of  cloth  is  divided  into  3  equal  pieces,  what 
part  of  a  yard  will  1  piece  contain  ? 

6.  By  what  do  you  divide  to  find  1-half  a  number?  To  find 
1-third?     1-fourth?     1-fifth? 

7.  How  do  you  multiply  by  |^  ?     By  J  ?     I^J  f  ? 

8.  What  is  the  difference  between  multiplying  by  J  and 
dividing  a  number  by  2?  Between  multiplying  by  |  and 
dividing  by  3  ? 

9.  If  5  fans  are  worth  i{-|,  what  is  1  fan  worth? 

10.  If  4  melons  cost  $i\-,  what  will  1  melon  cost? 


Division,  97 

11.  If  5  apples  are  worth  |  dime,  what  is  1  apple  worth? 

Analysis. — A  Fraction  is  divided  bv  dividing  its  numerator  or  multi- 
plying its  denominator.  Since  the  numerator  3  cannot  be  divided  by  5 
without  a  remainder,  we  multiply  the  denominator  4  by  it,  and  5  times  4 
are  20.     Therefore,  1  apple  is  worth  o%  dime,  Ans.     (Art.  179,  2".) 

12.  A  grocer  divided  f  of  a  cocoanut  among  6  boys  ;  what 
part  of  a  cocoanut  did  each  receive  ? 

13.  Paid  If  for  5  Table-books  ;  what  was  the  price  of  each  ? 

14.  How  many  ways  can  you  divide  a  fraction  ? 

15.  Divide  f  by  3.     ^  by  4.     -if  by  6.     f|  by  9. 

16.  Divide  |  by  3.     -J  by  4.     ^V  by  6.     H  by  5.     ^  by  H. 

17.  What  is  the  quotient  of  -^-^9  ?     Of  f^^S  ? 

18.  What  is  the  quotient  of  -?,6-  divided  by  9  ?     ^^..^n  p 

19.  A  man  had  ^  of  a  dollar,  and  gave  all  for  9  hats  ;  how 
mnch  did  each  hat  cost  him  ? 

20.  At  $4  a  bushel,  how  many  bushels  of  quinces  can  be  had 
for  llOf  ? 

Analysis. — As  many  busbels,  at  $4,  may  be  had,  as  $4  are  contained 
times  in  $10|.     Now  |10|  =  %^-,  and  %2.-^4  =  s^  or  2|  bushels,  Ans. 

21.  Divide  6|  by  4.     ^  by  5.     TJ  by  8.     11|  by  9. 

22.  If  8|  pounds  of  candy  are  divided  equally  among  5  chil- 
dren, what  part  and  how  much  will  each  receive  ? 

Written     Exercises. 
215.   Dividing  a  Fraction  by  an  Integer. 

1.  If  4  yds.  of  muslin  cost  $^,  what  will  1  yd.  cost  ? 

First. — Dividing  tbe  numerator  by  8-^-4 

4,  we  have  %f^  =  %\.     (Art.  179,  2\)  -T^'  =  ^TW  =  H^  -1^^'''' 


Second. — Multiplying    the  denom- 


:i 


inator  by  4,  we  have    — r — -  =  :^^,  or  x/c-  x  ■± 

1^  X  4:  iQ 

$t.  Ans.  ^ 

Or,  cancelling  the  factors  common  to  — :=  $^,    Aus. 

both  terms,  we  have  %\,  as  before.  « 

Notes. — 1.  It  is  better  to  divide  the  numerator  when  it  can  be  done 
without  a  remainder. 
5 


98  Frdctions, 

2.  By  the  first  operation  the  number  of  parts  is  diminished,  but  their 
size  remains  the  same.  Bj  the  second  operation  the  number  of  parts 
remains  the  same,  but  their  size  is  diminished. 


2.     Divide -V^  by  9.    6.     2||-^70=?        lO.     fo|-^-120=:? 

4.  H-12=:?  8.       J,V--^5:r.?  12.       iMf-75=? 

5.  f 


22=?  9.     3^-^-93  =  ?        13.     -i^V/--^14^=? 


14.  At  $7  a  barrel,  how  many  barrels  of  cranberries  can  be 
bought  for  $25|  ? 

Note. — When  the  dividend  is  a  mix-  operation. 

ed  number,  it  should  be  reduced  to  an  $25-|  =  %^-^ 

imj)roper    fraction;    then    proceed    as  |v?,  _l.  7  :=  |1J._  ^  $3%:. 
above.    Ans.  $3f. 

15.  If  2||  of  a  ton  of  hay  were  fed  to  6  horses,  what  part 
and  how  much  would  each  receive  ? 

16.  Paid  $7^  for  12  books  ;  what  was  the  price  of  each  ? 

Oral     Exercises. 

216.  1.  How  long  will  it  take  a  lad  to  earn  $5,  if  he  earns 
$1  a  day  ? 

Analysis. — At  $|  a  day,  it  will  take  as  many  days  as  f  are  contained 
times  in  5.  In  1  there  are  3  thirds,  and  in  5,  5  times  3,  or  15  thirds.  Now 
2  thirds  are  contained  in  15  thirds,  7^  times.     Aiis.  71  days. 

2.  How  many  times  is  J  contained  in  4  ?     In  5  ?     In  9  ? 

3.  How  many  times  1  in  5  ?     |-  in  7  ?    -J-  in  8  ?     |^  in  9  ? 

4.  If  you  earn  $J  in  1  day,  how  long  will  it  take  you  to 
earn  112  ? 

5.  How  many  times  f  in  7  ?     In  8?     In  10  ? 

6.  At  If  apiece,  how  many  books  can  you  buy  for  $12  ? 

7.  If  a  boy  saws  |  of  a  cord  of  wood  in  1  day,  how  long  will 
it  take  him  to  saw  8  cords  ? 

8.  How  many  times  are  f  contained  in  5  ?     In  G  ?     In  10  ? 

9.  At  $f  a  bushel,  how  much  corn  can  you  buy  for  110? 

10.  If  I  burn  f  ton  of  coal  in  1  day,  how  long  will  G  tons 
last  me  ? 

11.  How  many  times  are  f  contained  in  f  of  IG  ? 


Division.  .99 

12.  How  many  times  |  in  |  of  32  ?     In  f  of  40  ? 

13.  If  2|-  yards  of  cloth  will  make  a  coat,  how  many  coats 
can  be  made  from  20  yds.  of  cloth  ? 

Analysis. — In  2i  yards  there  are  5  half -yards,  and  in  20  yds.  there  are 
40  half-yards.     Now  5  is  contained  in  40,  8  times.     Ayis.  8  coats. 


14.  How  much  wood  at  $3^  a  cord  can  be  had  for 

15.  How   many   barrels   of   potatoes   at   $2f   can   you   buy 
for  $22? 

16.  At  $6f  a  week,  how  long  can  a  man  board  for  $100  ? 

Written     Exercises. 
217.   Dividing  an  Integer  by  a  Fraction. 

1.  How  many  times  are  f  contained  in  21  ? 

Explanation. — Reducing  21  to  fourths  opekation. 

we  have  21  =  V-     ^^ow  |  and  ^-  are  like  21   X  4  =  84 

fractions,  and  one  numerator  is  divided  by  ^_  _i_  |.  ;:^  28     Ans. 

the  other  like  integers.  r),    *,-•    ^   4  no 

Or,  Multiply  the  integer  by  the  fraction  ' 

inverted. 


2. 

56  by  ^\. 

5. 

240  by  A- 

8. 

384  by  if. 

3. 

72  by  A. 

6. 

256  by  ^-. 

9. 

576  by  if. 

4. 

132  by  11 

7. 

110  by  Y- 

10. 

1880  bv  /t 

11.  At  $^  a  yard,  how  many  j^ards  of  silk  can  be  had  for  $37  ? 

12.  If  you  pay  $J  a  day  for  board,  how  many  days  can  you 
board  for  $126  ? 

13.  How  many  cloaks  can  be  made  from  72  yds.  of  cloth, 
allowing  4 J  3'ds.  for  a  cloak  ? 

Note.— When  the  divisor  is  a  mixed  operation. 

number,  it  should  be  reduced  to  an  im-  4|^  =  |- 

proper  fraction    before    dividing ;    then  72  -h  #  =  16 

multiply  the  integer  by  the  fraction  in-  Qj.    tvo   y   2   _-  ;[g     Alls 
verted.    (Art.  217.  Ex.  \.)  '  "5  —       ^ 

14.  120-^12^=:?       16.     240-^yV=?      18.     785-^62izIr? 

15.  192-^10-|=r?       17.     552-^ff:=?      19.     2000-r-87i=? 

20.   At  $3^  apiece,  how  many  sheep  can  be  had  for  $1500  ? 


100  Fractions. 

21.  How  many  yards  of  silk,  at  %Z^  can  be  had  for  $185  ? 

22.  Allowing  4|  yards  of  cloth  for  a  cloak,  how  many  cloaks 
can  be  made  from  154  yards  ? 

23.  At  $4J  each,  how  many  chairs  can  be  bought  for  1250 
and  what  remainder. 

24.  If  a  stage  coach  travels  at  the  rate  of  lOf  miles  per 
hour,  how  long  will  it  be  in  going  320  miles  ? 

Oral    Exer  cises. 

218.     1.  How  many  slates  at  %-^  can  be  bought  for  $\^  ? 

Analysis. — Since  these  fractions  express  like  parts  of  like  units,  it  is 
plain  that  as  many  slates  can  be  bought  as  j\  are  contained  times  in  ^f, 
or  3.     Ans.  3  slates.    (Art.  197.) 

2.  If  a  vest  can  be  made  from  f  yd.  of  velvet,  how  many 
vests  can  be  made  from  ^  yards  ? 

3.  How  many  times  are  |  contained  in  ^-£-  ?    In  ^-^-  ?    In  -^^  ? 

4.  Divide  A  by  A.     ^  by  ^.    i|  by  j%.    J|  by  ^V 

5.  If  pen-knives  are  ||  apiece,  how  many  can  you  buy  for 

6.  How  many  melons  at  $|  apiece  can  a  person  buy  for  If  ? 

Analysis, — He  can  buy  as  many  as  |  are  contained  times  in  f .  Now 
I  =  f ,  and  I  are  cc-ntained  in  |,  2  times.     Ans.  2  melons. 

7.  How  many  books,  at  $f ,  can  be  bought  with  $|  ? 

8.  At  $f  a  yard,  how  much  fringe  will  If  buy  ? 

9.  At  if  a  pound,  how  many  pounds  of  spice  can  be  had 
for$^?     Forlfl? 

10.  Divide  f  by  A-     r\  by  f .    {%  by  ^.    U  by  i- 

11.  At  $J  a  yard,  how  many  vards  of  flannel  can  you  buy  for 

12.  How  many  pounds  of  tea  at  $f,  can  be  bought  for  $-|f  ? 

13.  At  $1,  how  many  yards  of  calico  can  be  had  for  S-j^  ? 

14.  If  cinnamon  is  $f  a  pound,  how  much  can  be  bought  for 

15.  How  much  coffee  can  be  bought  for  If  J  when  the  price 
is  If  a  pound  ?  - 


Division.  v^'",'^   \j      i(^i 

'     J  > 

'»©"     ••    jj 

Written    ExrR'cfscs. 
219.  Dividing  a  Fraction  by  a  Fraction. 

1.  How  much  tea,  at  $f  a  pound,  can  be  had  for  $|  ? 
1st  Method. — Reducing  the  given  frac-  1st  operation. 

tions  to  a  c.  d,,  |  =  j^,  and  f  =  j%     Now  if         |  =  -^  ;     f  =  i^^ 
$^%  will  buy  1  pound,  $j%  will  buy  as  many  ^9^  _^  _8_  —  9  _i_  g 

pounds  as  y%  are  contained  times  in  y\,  and       g  _j_  g  __  -j^  1  2|)_     AflS^ 
9h-8  =  1|.  "^ws.  Impounds.     (Art.  191.)  '  8       •?  * 

2d  Method.— The  above  process  may  be  ~°  opekation. 

shortened  by  inverting  the  divisor  and  mul-  $|-  -^  §-|  =:  f  X  ^ 

tiplying  the  two  fractions  together  as  in  the      |-  x   |  =  -§,  or  1^  lb. 
margin.     (Art.  210.) 

Note. — 1.  It  will  be  seen  by  inspection  that  the  2d  method  in  effect 
reduces  the  fraction  to  a  c.  d.  and  divides  one  numerator  by  the  other  at 
the  same  time,  the  numerators  only  being  used,  as  in  1st  method. 

2.  Divide  10|  by  6|. 

Solution.  —  Reducing     the  operation. 

mixed    numbers,    to    improper  10|-  =  ^^-  ;      6|-  ==  -^ 

fractions  and  dividing,  the  result  8j_  _^  2_i  =:  JUL  x  -A-  =  14,  Ans. 
is  14. 

3.  Divide  J  of  f  by  4  of  -jV 

4 
Solution.— I  X  |  X  |  X  -^^  =  f,  or  IJ,  Ans. 
3 

220.  The  preceding  principles  may  be  summed  up  in  the 
following 

General     Rule. 

Reduce  iclwle  and  mixed  numbers  to  improper  frac- 
tions, and  multiply  the  dividend  by  the  divisor  inverted. 

Or,  Reduce  the  fractions  to  a  commoii  denominator 
and  divide  the  numerator  of  the  dividend  by  that  of  the 
divisor. 

Note. — The  object  of  inverting  the  divisor  is  convenience  in  multiply- 
ing.    After  inverting  the  divisor,  cancel  the  common  factors. 

4.  Divide  -f^  by  \.  6.  Divide  81|  by  45|. 

5.  Divide  75  by  Sf.  7.  Divide  i^j  of  f  by  30. 


102  J^r  actions. 


\^r  PLICATIONS. 


221.     1.  At  16J  cents  per  pound,  how  many  pounds  of  figs 

can  you  buy  for  87|-  cents  ? 

2.  How  many  cords  of  wood,  at  $6J  per  cord,  will  it  take  to 
pay  a  debt  of  I67J-  ? 

3.  How  many  barrels  of  pork,  at  $llf  per  barrel,  can  be 


obtained  for  $95J  ? 


4.  A  man  bought  15J  barrels  of  beef  for  $124| ;  how  much 
did  he  give  per  barrel  ? 

5.  A  man  bought  13 1  pounds  of  sugar  for  94r|-  cents ;  how 
much  did  his  sugar  cost  him  a  pound  ? 

6.  A  lady  bought  15|-  yards  of  silk  for  IIS^^^  shillings  ;  how 
much  did  she  pay  per  yard? 

7.  Bought  15^  baskets  of  peaches  for  |24|  ;  how  much  was 
the  cost  per  basket  ? 

8.  Bought  30:^  yards  of  broadcloth  for  |181|- ;  what  was  the 
price  per  yard  ? 

9.  Paid  $375  for  l'2b\  pounds  of  indigo  ;  what  was  the  cost 
per  pound  ? 

10.  How  many  tons  of  hay,  at  IIGJ  per  ton,  can  be  bought 
for  $1961? 

11.  How  many  sacks  of  wool,  at  $17^  per  sack,  can  be  pur- 
chased for  $1500  ? 

12.  How  many  bales  of  cotton,  at  II 5|  per  bale,  can  be 
bought  for  12500  ? 

Divide  ^  of  16  by  |  of  |. 
Divide  f  of  |  by  21. 
21.  Divide  y\  of  -^  by  I  of  f 
Divide  223-J  by  f  o/si. 
Divide  |  of  |  by  48. 
Divide  42J  by  |  of  53^ 

25.  Divide  J  of  }  of  -^  of  f  of  -Jf  by  |  of  if. 

26.  Divide  y^  o^  f  o^  tI  of  ^  by  -|f  of  \\  of  18. 

27.  Divide  -|f  of  if  of  67  by  |J  of  f|  of  25. 

28.  Divide  ff  of  |^  of  4U  by  ff  of  |i  of  31. 

29.  Divide  fj  of  |4  of  |f  "of  82f  by  \\  of  f|  of  42f. 


13. 

Divide  |  of  y^  by  6f 

19. 

14. 

Divide  ^V  of  30  by  19. 

20. 

15. 

Divide  i\  of  1%  by  31. 

21. 

16. 

Divide  ^  by  |  of  12. 

22. 

17. 

Divide  ^  by  ISf 

23. 

18. 

Divide  42^  by  |  of  7. 

24. 

Division.  *  103 

222.   To  Reduce  Complex  Fractions  to  Simple  Ones. 

Expressions   which   have   a   Fraction   in   the  numerator  or 
denominator  or  in  both,  are  called  Complex  Fractions. 


ing  Division  of  Fractions. 


Thus,  if  ;  57  ;  if  ;  f  >  are  complex  fractions,  and  are  a  form  of  indicat- 


1.  What  is  the  yalue  of  T-f- 


OPEBATION. 


Analysis. — Reducing  the  mixed   numbers  to  31   -—  10 

improper  fractions,  we  divide  the  numerator  by  03  40 

the  denominator  according  to  the  rule.    (Art.  220.)  10.48  5_o 

The  result  is  AOx  =  #1,  Ans.    Hence,  the  -X"  ~  ^"  —  m 

EuLE. — Treat  the  numerator  as  a  dividend  and  the 
denominator  as  a  divisor,  and  divide  one  by  the  other 
according  to  the  rule  for  division  of  fractions. 

2.  Reduce  ^  to  a  simple  fraction.     Ans.  ff. 

Reduce  the  following  to  their  simplest  form  : 

6  ^      121  94  251 

3.  — ■•  6.      — -•  9.      — -.  12. 

H  6i  7J  f 

4.  -•  7.     -^-  10.      — -.  13.      — . 
■I                           6                           12J  li 

8  44  20|  « 

Note. — Complex  Fractions,  when  reduced  to  Simple  Fractions,  are 
added,  subtracted,  multiplied,  and  divided  like  other  fractions. 

15.  Find  the  sum  of  the  2d  and  3d. 

16.  Find  the  difference  of  the  4th  and  5th. 

17.  What  is  the  product  of  the  6th  by  the  7th  ? 

18.  What  is  the  quotient  of  the  10th  divided  by  the  9th  ? 

19.  What  is  the  product  of  the  7th  and  8th  ? 

20.  What  is  the  quotient  of  the  12th  diWded  by  the  13th  ? 


104  -  Fractions, 

223.  Finding  what  Part  one  Number  is  of  Another. 

1.  What  part  of  9  inches  is  2  inches  ?    4  in.  ?     7  in.  ? 

2.  What  part  of  a  yard  is  1  foot  ? 

Analysis. — In  1  yard  there  are  3  feet,  and  1  foot  is  \  of  3  feet,  Ans. 

3.  What  part  of  a  week  is  1  day  ?     2  days  ?     5  days  ? 

4.  What  part  of  3  days  is  1  foot  ? 

Ans.  Days  and  feet  are  unlihe  numbers,  and  therefore  one 
cannot  be  compared  with  the  other. 

5.  What  part  of  -f  is  f  ? 

Analysis. — Since  these  fractions  have  a  c,  d,  they  are  like  fractions, 
and  their  numerators  are  compared  like  integers.     Ans.  f . 

224.  From  the  examples  above  are  derived  the  following 

Principles. 

1°.  Only  like  mimhers,  or  those  wliich  are  so  far  of  the 
same  hind  that  one  may  he  said  to  he  a  part  of  the  other, 
can  he  compared. 

2°.  When  fractions  have  a  comynon  de7iominator,  their  nu^ 
merators  are  compared  lihe  integers. 

Oral     Exercises. 

225.  1.  What  part  of  30  cents  are  5  cents  ?    Ans.  -^q  or  -J-. 

2.  What  part  of  21  yards  are  7  yards?  Of  45  days  are 
9  days  ? 

3.  $7  are  what  part  of  $15-?     Of  $45  ?     Of  $63  ? 

Find  what  part  one  of  the  following  numbers  is  of  the  other, 
expressed  in  lowest  terms : 

4.  Of  30  is  12  ?  6.   Of  96  is  48  ?  8.   Of  65  is  100  ? 

5.  Of  63  is  14  ?  7.   Of  120  is  30  ?         9.  Of  108  is  144  ? 

10.  If  an  acre  of  land  is  worth  $63,  what  part  of  an  acre  will 
$9  buy  ? 

11.  If  a  piece  of  carpeting  can  be  bought  for  $120,  what 
part  of  a  piece  can  be  bought  for  $12  ? 

12.  23  is  what  part  of  69  ?     48  of  72  ?     84  of  99  ? 


Division.  105 

13.  I  is  what  part  of  ^  ?    t%  of  H  ?    H  ^f  fj  ? 

14.  AYhat  part  of  ^  is  y^^-  ? 
Suggestion.— I  =  -f^.    Ans.  |. 

15.  What  part  of -H  is  f  ?     Of|is-|^? 

16.  What  part  of  ^  is  iV  ? 

Written     Exercises. 
226.  To  find  what  part  one  number  is  of  another. 

1.  What  part  of  49  is  28  ? 
Analysis. — 28  is  ff  of  49,  or  |  of  49,  Ans. 

2.  What  part  of  tV  is  J-g-  ? 

Analysis. — Reduced  to  a  c.  (J,  the  given   frac- 
tions become  |f  and  f f ,  which   are  like  fractions.  22 

Now  22  is  II  of  35,  Ans.    (Art.  224.)     Hence,  the  so  —  6 0 

22  -^  35  =  II 

Rule.  —  Mahe  the  Jiimider  denoting  the  pa?'t  the 
numerator,  and  that  with  luhieh  it  is  compared  the 
denominator. 

Note. — If  either  or  both  the  given  numbers  are  fractional,  they  should 
be  reduced  to  a  c,  d,  :  their  numerators  are  then  compared  like  integers. 


OPERATIOK. 

T¥  —    6  0 


5 


3.  What  part  of  36  is  -|  ?  9.   lOOf  is  what  part  of  175f 

4.  What  part  of  62  is  -J  ?  10.   6 J  is  what  part  of  45  ? 

5.  What  part  of  86  i^  |i  ?  11.  40  is  what  part  of  954  ? 

6.  AYhat  part  of  58  is  7f  ?  12.  if  is  what  part  of  -Jg.  ? 

7.  What  part  of  112  is  |  ?  13.  |f  is  what  part  of  ^  ? 

8.  What  part  of  325  is  I  ?  14.   18|-  is  what  part  of  46f  ? 

15.  At  S23  per  acre,  how  much  land  will  $17  buy  ? 

16.  A  man  paid  $185  for  a  horse,  and  sold  it  for  $150 ;  what 
part  of  the  cost  did  he  get  ? 

17.  A  man  76  years  old  has  a  son  whose  age  is  54  years ; 
what  part  of  the  father's  age  is  that  of  his  son  ? 

18.  If  from  a  piece  of  silk  coutaining   27j-  yds.,  you   cut 
11|^  yds.,  what  part  of  the  pie,ce  will  be  left  ? 


106  Fractions. 

19.  If  a  man  can  perform  a  journey  in  24  days,  what  part 
of  it  can  he  go  in  9  days  ? 

20.  What  part  of  |268|  is  Il75f  ? 

21.  If  A  can  do  a  job  of  work  in  20  days,  and  B  in  10  days, 
what  part  will  each  do  in  1  day  ?     What  part  will  both  do  ? 

Oral     Exercises. 

227.  1.  4  is  4"  of  what  number  ? 

Analysis.— 4  is  -|  of  4  times  7,  or  28.    Therefore,  4  is  |  of  28,  Ans. 

2.  36  is  f  of  what  number  ? 

Analysis.— Since  36  is  |  of  a  number,  |  of  tliat  number  is  ^of36, 
which  is  12,  and  4  fourths  are  4  times  12,  or  48.     Therefore,  etc. 

Note. — If  the  learner  is  at  a  loss  which  term  of  the  fraction  to  take  for 
the  divisor,  let  him  substitute  the  word  i^arts  for  the  denominator,  and  his 
difficulty  will  vanish. 

3.  15  is  I  of  what  ?  7.  15|  is  f  of  what  ? 

4.  16  is  f  of  what  ?  8.  10|  is  4  of  what  ? 

5.  45  is  4  of  what  ?  9.  45  ==  |^f  of  what  ? 

6.  210  is  I  of  what  ?  10.  48  =  i|  of  what  ? 

Written    Exercises. 

228.  To  find  a  Number  when  a  Part  of  it  is  given. 

1.  56  is  J  of  what  number? 

.  CI-  1        £  \  •        t;a  OPERATION. 

Analysis.  —  Since  f  of  a  number  is  56, 
1  ninth  is  \  of  56,  which  is  8,  and  9  ninths  are         Ob  -r-  /  =  o 
9  times  8,  or  72,  A7is,     Hence,  the  8  X  9  r=  72,  Ans. 

KuLE. — Divide  the  numher  denoting  the  part  hy  the 
ninneratoi%  and  multiply  the  quotient  by  the  denomi- 
nator.    (Art.  208.) 

2.  48  is  I  of  what  ?  6.  132  is  f|  of  what  ? 

3.  56  is  I  of  what  ?  7.  257  is  f  of  what  ? 

4.  75  is  f  of  what  ?  8.  394  is  ^  of  what  ? 

5.  96  is  T^o  of  what  ?  9.  859  is  \l  of  what  ? 

10.  A  merchant  lost  $4368,  which  was  ^  of  his  capital  ; 
what  was  his  capital  ? 


Division.  107 

11.  If  f  of  a  farm  is  worth  $2360,  what  is  the  whole  worth  ? 

12.  A  drover  being  asked  how  many  sheep  he  had  replied 
that  147  was  equal  to  ^^  of  them  ;  how  many  sheep  had  he  ? 

13.  A  man  lost  f  of  his  money  and  had  $260  left ;  how  much 
had  he  at  first  V 

Oral    Problems    for    Review. 

229.  1.  A  lad  having  $5,  paid  $2|^  for  a  pair  of  skates  and 
$1^  for  a  sled  ;  how  much  did  he  have  left  ? 

2.  A  lady  went  shopping  with  $15  in  her  purse  ;  she  paid 
$f  for  a  handkerchief,  S2^  for  a  pair  of  gloves,  and  the  rest  for 
a  shawl ;  what  did  the  shawl  cost  her  ? 

3.  A  laborer  earned  $1J  one  day,  $1^  the  next,  and  paid  $1|- 
for  board  ;  how  much  had  he  left  ? 

4.  A  grocer  bought  a  load  of  apples  at  $|  a  bushel,  and  sold 
them  at  $f ;  how  much  did  he  make  on  a  bushel  ? 

5.  A  man  owning  ^  of  a  ship,  sold  f  of  her ;  what  part  had 
he  left? 

6.  The  sum  of  two  fractions  is  ^^  and  one  of  them  is  f ; 
what  is  the  other  ?     What  is  their  difference  ? 

7.  The  greater  of  two  numbers  is  6f,  and  their  difference 
is  2J  ;  what  is  the  less  number  ? 

8.  The  less  of  two  numbers  is  5|,  and  their  difference  2-|- ; 
what  is  the  greater  number  ? 

9.  The  product  of  two  fractions  is  \\,  and  one  of  the  frac- 
tions is  J  ;  what  is  the  other  fraction  ? 

10.  If  the  dividend  is  ^,  and  the  quotient  is  -f,  what  is  the 
divisor  ? 

11.  What  number  divided  by  |  will  give  a  quotient  of  7^  ? 

12.  A  teacher  spends  f  of  his  salary  for  board  and  -^^  for 
clothing  ;  what  part  of  his  salary  is  left  ? 

13.  If  a  man  earns  $60  a  month  and  spends  f  of  it,  how 
much  can  he  lay  up  ? 

14.  I  sold  I  of  my  farm  and  had  48  acres  left ;  how  many 
acres  did  my  farm  contain  ? 

15.  What  is  the  difference  between  4|^  and  o|  ? 

16.  What  number  subtracted  from  15|-  will  leave  10^%? 


108  ^'r  actions. 

17.  The  sum  of  two  fractions  is  17|,  and  one  of  them  is  12f ; 
what  is  the  other  ? 

18.  At  %VZ^  a  sack,  what  are  5  sacks  of  coffee  worth  ? 

19.  At  $5f  a  barrel,  what  will  10  barrels  of  flour  cost? 

20.  At  $6  a  ton,  what  will  15 1  tons  of  coal  ceme  to? 

21.  What  will  9  cords  of  wood  cost,  at  1^3 J  a  cord  ? 

22.  Bought  a  horse  and  sleigh  for  1175,  and  the  sleigh  was 
worth  f  as  much  as  the  horse  ;  what  was  the  value  of  each  ? 

23.  A  lady  bought  6  neck-ties  at  S|  each,  aud  has  $15  left ; 
how  much  money  had  she  at  first  ? 

Written    Problems    for    Review. 

230.     1.  Reduce  ^  to  the  denominator  243. 

2.  Reduce  ||||  to  lowest  terms. 

3.  Find  the  prime  factors  of  486,  576,  and  972. 

4.  What  is  the  I,  c,  d»  of  ~-f-^,  |,  and  J|-  ? 

5.  What  is  the  sum  of  f  of  f ,  i  |,  and  5|-  ? 

6.  What  is  the  difference  between  14i  +  25f,  and  25f +  19i? 

7.  The  greater  of  two  numbers  is  375|,  and  their  difference 
273f  ;  what  is  the  less  ? 

8.  If  I  buy  H  ^f  ^  ship,  and  sell  |  of  what  I  buy,  how  much 
shall  I  then  own  ? 

9.  Required  the  sum  and  difference  of  }  and  ^  ? 

10.  A  railroad  car  goes  225J  miles  in  a  day  and  a  steamer 
185J  miles  ;  how  far  do  both  go  in  a  day,  and  what  is  the  dif- 
ference m  the  distance  they  go  ? 

11.  A  grocer  bought  a  quantity  of  apples  for  $162|-  and  sold 
them  for  |210f  ;  what  was  his  profit? 

Find  the  sum  of  the  following  : 

(24|  +  12i)-(llJ  +  2f). 
(28-2f)  +  (16~2A). 
(140  +  l|-)-(8A-li). 
145  +  ^V+(112|-8t). 

20.  A  farmer  sold  a  cow  for  $26f,  15  sheep  for  $52J,  and 
the  buyer  handed  him  a  $100  bill  ;  how  much  change  should 
he  return  ? 


12. 

8i-f6^-3i. 

16. 

13. 

14|4-6i-7f. 

17. 

14. 

20f-8i-  +  4|. 

18. 

15. 

26f  +  (4}-2i)+3i. 

19. 

Division.  109 

21.  Paid  $275f  for  a  quantity  of  wheat,  $320^  for  a  quan- 
tity of  corn,  and  sold  the  former  for  I316|,  and  the  latter  for 
$41 0|- ;  what  w\as  my  profit  ? 

22.  If  t  of  a  factory  cost  123245,  what  is  the  whole  worth  ? 

23.  What  number  multiplied  by  7f  will  produce  872|? 

24.  If  the  divisor  is  f^,  and  the  quotient  ^,  what  must  be 
the  dividend? 

25.  The  dividend  is  42f,  the  quotient  8^,  what  is  the  divisor? 

26.  A  father  bequeathed  J,  \,  and  ^  of  his  property  to  his 
3  children,  and  had  14800  left  for  his  wife  ;  what  was  the 
amount  of  his  property  ? 

27.  A  merchant  lost  f  of  his  capital  by  one  creditor,  and  f 
by  another,  and  had  1500  left  ;  what  was  his  capital  ? 

28.  My  neighbor  having  356|  acres  of  land,  sold  ^  of  it  to 
one  man,  and  f  of  it  to  another";  what  was  the  value  of  the 
remainder  at  I25f  per  acre  ? 

29.  A  man  gave  his  check  for  $1675J,  which  was  |  of  what 
he  had  on  deposit ;  how  much  had  he  in  bank  ? 

30.  A  man  had  6f  acres  of  land,  which  he  divided  into 
building  lots  each  containing  ■^\  acres  ;  how  many  lots  did 
he  have  ? 

31.  Bought  a  horse  for  $160J,  and  sold  it  for  f  of  the  cost ; 
how  much  did  I  lose  ? 

32.  How  many  books,  at  $-|  apiece,  can  be  bought  for  810-^  ? 

33.  At  13 J  a  day,  how  much  can  a  man  earn  in  25|-  days  ? 

34.  What  cost  34J  bushels  of  flaxseed,  at  $2^  a  bushel  ? 

35.  A  market-woman  being  asked  how  many  eggs  she  had, 
replied,  "  244  equals  f  of  them  ; "  how  many  had  she  ? 

36.  A  man  paid  $5250  for  a  house,  which  was  -f-  of  all  his 
property ;  how  much  was  he  worth  ? 

37.  If  T^  of  a  ship  cost  S8360,  what  is  the  whole  ship  worth  ? 

38.  A  lady  teacher  paid  $750  for  a  piano,  which  was  f  of 
her  salary  for  a  year ;  what  was  her  salary  ? 

39.  A  tree  casts  a  shadow^  of  48  feet,  which  is  f  of  its  height ; 
how"  high  is  the  tree  ? 

40.  Nine  feet  of  a  flag-staff  stands  in  the  ground,  which  is 
■^  of  its  w^hole  length ;  what  is  its  length  ? 


110  Fractions. 

41.  A  lad  being  asked  how  many  marbles  he  had,  said  he 
had  f  as  many  as  his  friend,  and  that  both  together  had  255  ; 
how  many  had  he  ? 

42.  A  goldsmith  paid  175  for  a  watch,  which  was  f  of  what 
he  got  for  it ;  how  mnch  did  he  get  for  the  watch  ? 

43.  A  can  build  a  school-house  in  90  days,  which  is  f  of  the 
time  it  would  take  C  ;  how  long  would  it  take  C  to  build  it  ? 

44.  An  army  lost  \  of  its  men  in  battle  and  \  by  sickness, 
and  had  9600  left ;  what  was  its  whole  number  ? 

45.  16|  is  -i-  of  what  ?      48.     f  of  G|  is  f  of  what  ? 

46.  18|  is  ^  of  what  ?    49.     |  of  ff  is  ^^  of  what  ? 

47.  25f  is  f  of  what  ?       50.     f  of  48  is  how  many  times  10  ? 

51.  A  man  bought  a  buggy  for  §185,  which  was  f  the  price 
of  his  horses ;  what  did  his  horses  cost  ? 

52.  A  whale-ship  lost  -^-^  of  the  bread,  and  the  men  were 
allowed  12  ounces  per  day  a2:)iece  ;  what  had  each  at  first  ? 

53.  A  man  sold  his  farm  for  $4760,  and  thereby  gained  J  of 
the  cost ;  what  did  he  pay  for  it  ? 

54.  A  man  bequeathed  to  his  son  17600,  which  was  If  of  what 
he  gave  his  daughter;  what  was  his  daughter's  portion  ? 

Questions.  *^ 

160.  What  is  a  fraction?  161.  The  unit  of  a  fraction?  162.  A  frac- 
tional unit?    164.  What  is  the  denominator  ?     165.  The  numerator  ? 

166.  What  are  the  terms  of  a  fraction?  171.  What  is  a  proper  frac- 
tion? 172.  Improper?  173.  Simple?  174.  A  Compound?  175.  A 
mixed  number  ? 

176.  From  what  do  fractious  arise?  177.  What  is  the  value  of  a  frac- 
tion ?     179.  Name  the  three  general  principles  of  fractions. 

181.  What  is  reduction  of  fractions?  182.  How  is  a  fraction  reduced  to 
higher  terms?  185.  How  to  the  lowest  tei-ms ?  187.  Improper  fractions 
to  mixed  numbers  ?     189.  Mixed  numbers  to  improper  fractions? 

191.  What  is  a  common  denominator?  192.  The  least  common  denom- 
inator? 196.  How  found  ?  197.  What  are  like  fractions?  198.  Unlike? 
200.  What  fractions  can  be  added?  201.  Rule  for  adding  fractions? 
204.  Rule  for  subtracting  fractions  ? 

211.  The  force  of  the  word  "of"  in  compound  fractions  ?  212.  General 
rule  for  multiplying  fractions?     220.  General  rule  for  dividing  fractions? 

222.  What  are  complex  fractions  ?  How  reduce  complex  fractions 
to  simple  ones?  226.  How  find  what  part  one  number  is  of  another? 
228.  How  find  ^  number  when  a  part  of  it  is  given  ? 


f        I  »    ... 9 

1 — ■ —  I     g)  !  ^^"'    i— 

ECIMAL    FkACTIOIsTS. 


-i=^.\~/l=r 


Oral     Exercises. 

231.  1.  If  a  unit  is  divided  into  10  equal  parts,  what  is 
each  part  called  ? 

2.  If  one  of  these  tentlis  is  subdivided  into  10  equal  parts, 
what  part  of  the  unit  is  one  of  them  ? 

Ans.  TS"  ^^  iV  —  Too"?  or  one  hundredth. 

3.  What  part  of  the  unit  is  2  of  these  parts  ?  3  of  them  ? 
6  of  them?     11  of  them? 

4.  If  1  hundredth  of  a  dollar  is  divided  into  10  equal  parts, 
what  part  of  a  dollar  is  one  of  these  parts  ? 

Ans.   iV  of  To-o  =  roVo?  or  one-thousandth. 

5.  What  part  of  a  dollar  is  2  of  these  parts  ?     4  parts  ? 

6.  What  is  meant  by  a  tenth  ?    3  tenths  ?     7  tenths  ? 

7.  What  is  meant  by  a  hundredth  9    4  hundredths  ? 

8.  What  is  meant  by  a  thousandth  9    5  thousandths  ? 

Definitions. 

232.  A  Decimal  Fraction  is  one  or  more  of  the  equal  parts 
of  a  unit  divided  into  tenths,  hundredths,  thousandths,  etc. 

Note. — They  are  called  Decimals  from  the  Latin  decern,  ten,  which 
indicates  their  origin  and  scale  of  decrease. 

233.  A  Mixed  Decimal  is  an  integer  and  decimal  expressed 
together. 

Thus,  34.153,  and  42.65  are  mixed  decimals. 

234.  Decimals  are  expressed  by  writing  the  numerator  only, 
with  a  decimal  point  ( . )  on  the  left. 

235.  The  Denominator  of  a  decimal  is  always  10,  100,  1000; 
etc. ;  or  1  with  as  many  ciphers  annexed  to  it  as  there  are 
decimal  places  in  the  given  numerator. 


112 


Decimal  Fractions, 


236.  The   Notation   of   Decimals  is   an   extension   of    the 
Notation  of  Integers.     (Art.  36.) 


Table. 

'!3 

• 

m 

fl 

X! 

^ 

(3 

aj 

-M 

OQ 

_o 

OB 

aj 

-a 

J3 

Names 

s 

<J0 

O 

T3 

a 

a 

a 

03 

a 

o 

'SI 

s:3 

.=3 

S 

• 

^i 

CQ 

aJ 

_o 

of 
Ordei's, 

.2 
1 

-.J 
o 

O 

-2 

00 

^3 

8 

a 

08 

a 

o 

.a 

OB 

_o 

:;:; 

o 

0 

Si 

a 

Cm 

o 
a 

0) 

-a 

a 
a 

as 

s    a 

QQ 

.a 
a 

0) 

1 
-a 
a 

a 

e 

o 
a 

a 

0 

a 

a 
a 

n 

Er- 

W 

E^ 

h 

ffi 

e^   P 

Eh 

W 

H 

e< 

W 

H 

W 

Number. 

6 

3 

8 

,  4 

2 

5 

,    6 

7'2   . 

3 

2 

,  6 

7 

2   , 

5 

4 

5 

Orders. 

ji 

^ 

^' 

j= 

^' 

J* 

'3 

T3       "S 

-d 

^ 

.d 

^ 

-4^ 

.a' 

5* 

o 

00 

i^ 

s 

io 

^ 

CO 

C*         r-l 

<N 

CO 

■^ 

lO 

s 

I- 

00 

TO 

v~~" 

V                    ~ 

Integers. 

Decimals. 

The  number  is  read,  Six  hundred  thirty-eiglit  millions,  four 
hundred  twenty-five  thousands,  six  hundred  seventy-two,  and 
thirty-two  million  six  hundred  seventy-two  thousand,  five  hun- 
dred forty-five  liundred-millionths. 

The  scale  of  decrease  of  decimal  orders  may  be  illustrated 
by  the  following  diagram: 


_i 

10  0  0 


237.  The  value  of  each  figure  in  decimals,  as  well  as  in 


integers,   is  determined  by  the  place 
from  units. 


it   occupies,  counting 


Thus,  a  figure  in  tlie  first  place  on  the  right  of  the  decimal  point,  has 
ten  times  the  value  of  the  same  figure  in  the  next  lower  order,  or  hun- 
dredths place,  and  only  one-tenth  the  value  it  would  have  in  units  place. 


Notation.  113 

238.  Orders  equally  distant  on  the  riylit  and  left  from  units 
place,  have  corresponding  names.  Thus,  tenths  corres^Dond  to 
tens,  hundredths  to  hundreds,  etc.     Hence, 

239.  The  Numerator  of  a  decimal  fraction,  when  written 
alone,  must  contain  as  many  figures  as  there  are  ciphers  in  its 
denominator.  If  it  has  not  significant  figures  enough,  the 
deficiency  must  be  supplied  by  prefixing  ciphers. 

Thus,  yI^  expressed  decimally  is  .05  ;  -^-^.^^  is  .005,  etc.     Hence, 

240.  To  lurite  decimals,  we  have  the  following 

Rule. —  Write  tenths  in  the  first  decimal  place,  hun- 
dredths i?i  the  second,  thousandths  in  the  third,  etc. 

Write  the  following  fractions  decimally :     (Art.  239.) 

1. 
2. 
3. 

13.  Forty-two  hundredths.        16.  43  ten-thousandths. 

14.  Twentj^-one  thousandths.    17.   65  hundred-thousandths. 

15.  Six  ten-thousandths.  18.  426  millionths. 

19.  18^.  22.      60^U^.  25.      SStoVw 

20.  23y^.  23.      lOO^J^.  26.      6^jUh' 

21.  28Ttto-  24.     243tL2_5_.  27.     93^1^. 

241.  To  Read  Decimals  expressed  by  Figures. 

1.  Read  the  decimal  0.000427. 

Explanation. — Beginning  at  units,  we  say,  "  units,  tenths,  hundredths, 
thousandths,"  etc.,  to  the  lowest  order,  which  is  millionths.  We  now  read 
the  significant  figures  as  if  integers,  and  pronounce  the  name  milliorUha. 
Am.  Four  hundred  twenty-seven  millionths.     Hence,  the 

Rule. — Read  the  significant  figures  of  the  decimal  as 
integers,  and  give  it  the  name  of  the  lowest  order. 

Notes. — 1 .  In  mixed  decimals,  read  the  integral  part  as  if  it  stood  alone, 
then  read  the  decimal.  Or,  ha\ing  read  the  integral  part,  pronounce  the 
word  "decimal"  then  read  the  decimal  figures  as  if  integers. 


1 

1  0- 

4. 

5 

100  0* 

7. 

ioo\o- 

10. 

TOOO  00* 

10  0"* 

5. 

82 
10  0  0* 

8. 

204 
100  00* 

11. 

100000' 

l^^o. 

6. 

10.5 
lOOlT' 

9. 

506 

10  0  0  0* 

12. 

6803 
100000- 

114 


Decimal  Fractions, 


2.  In  reading  mixed  decimals,  the  word  "and"  should  not  be  iised 
except  between  integers  and  decimals. 

242.  Copy  and  read  the  following  : 


2. 

.07. 

10. 

7.042. 

18. 

.0072. 

3. 

.005. 

11. 

16.0039. 

19. 

.00201. 

4. 

.102. 

12. 

23.0142. 

20. 

.400025. 

5. 

.0624. 

13. 

62.00301. 

21. 

.000367. 

6. 

.00206. 

14. 

73.04605. 

22. 

6.043216. 

7. 

.024542. 

15. 

8.20304. 

23. 

45.002064. 

8. 

.000821.       ^ 

16. 

68.207308. 

24. 

.00004607. 

9. 

.0000265. 

17. 

95.000206. 

25. 

.020605027 

Reduction    of    Decimals. 

Oral     Exercises. 

243.  1.  How  many  tenths  in  1  ?     How  many  hundredths  ? 
How  many  thousandths  ? 

2.  How  many  tenths  in  2  ?     Tn  5  ?    In  6  ? 

3.  How  many  hundredths  in  3  ?     In  4  ?     In  7  ? 

4.  How  many  thousandths  in  4  ?     In  6  ?     In  8? 

5.  How  many  thousandths  in  5  ?    In  7  ?     In  9  ? 

Illustration  of  Pbincii^les. 

244.  Since  the  orders  of  decimals  decrease  from  left  to  riglit 
by  Tens,  it  follows  : 

i°.  Prefixijig  a   ciplier  to  a  decimal,  diininishes   its  value 
10  times,  and  reduces  it  to  tlie  next  loiver  order. 
Thus,  .5  =  /o  ;  but  .05  =  y§o ;  -005  =  yxfV o.  etc. 

^°.  Removing  a  cipher  from  the  left  of  a  decimal,  increases 
its  value  10  times,  and  reduces  it  to  the  next  higher  order. 


Thus,  .005  =  y^oT)  ;  l»^t  -^5 


TIT 


,  etc. 


5°.  A7inexing  a  cipher  to  a  decimal,  or  re?noving  one  from  its 
right,  does  not  change  its  value. 

Thus,  .5  =  ^^  ;  also  .50  =  -j^JV  ;  -500  =  j%%%,  all  of  which  are  equal. 


Reduction.  115 

Written     Exercises. 

245.  To  Reduce  Decimals  to  a  Common  Denominator. 

1.  Eeduce  .5,  .42,  and  .006,  to  a  common  denominator. 

Analysis. — The  lowest  order  of  the  given  decimals  operation. 

is  thousandths.      Annexing  ciphers  to  decimals  does  .5  =  0.500 

not  change  their  value.     The  fractions  are  .500,  .420,  .42  ==:  0.420 

and  .006,  J.?i«.     (Art.  244,  5°. )     Hence,  the  .006  =  0.006 

Rule. — Annex  to  each  as  many  ciphers  as  may  he  re- 
quired to  mahe  their  decimal  places  equal. 

2.  Reduce  .20,  2.0004,  and  7.008,  to  a  c.  d. 

3.  Reduce  2  tenths,  6  hundredths,  and  8  thousandths,  to  a 
common  denominator. 

4.  Reduce  .03,  .125,  .7,  and  .2362,  to  a  c.  d, 

5.  Reduce  .26,  .275,  .0236,  and  .206,  to  a  c.  d. 

6.  Reduce  .045,  .61,  .0035,  and  .108,  to  a  c,  d. 

Oral     Exercises. 

246.  1.  Reduce  .5  to  a  common  fraction. 

Analysis. — 0.5  =  ^^,  and  -^^  reduced  to  its  lowest  terms,  equals  \,  An» 

2.  How  many  halves  in  .50  ?     In  .500? 

3.  How  many  fifths  in  .  4  ?     In  .  6  ? 

4.  How  many  fourths  in  .25  ?     Fifths  in  .20  ? 

5.  How  many  tenths  in  .40  ?     In  .60  ? 

6.  How  many  twentieths  in  .60  ?     In  .80  ? 

Written     Exercises. 

247.  To  Reduce  Decimals  to  Common  Fractions. 

1.  Reduce  .68  to  a  common  fraction. 

Solution.— The  denominator  of  .68  is  100.  Therefore,  .68  =  ^^ 
or  ^|.     (Art.  235.)     Hence,  the 

Rule. — Erase  the  decimal  point,  write  the  numerator 
over  its  denomijiator .  and  reduce  the  fraction  to  its 
lowest  terms.     (Art.  185.) 


116  Decimals. 

2.  Eeduce  .33 J  to  a  common  fraction  in  the  lowest  terms. 
33-t 


A71S.  .33-1-  , 

_   '^'^  3    _    10  0 
-    -^QQ   -    300. 

or 

h 

Eeduce  the 

following : 

3.     0.28. 

7.     0.05. 

11. 

0.005. 

15. 

0.410007. 

4.     0.56. 

8.     0.008. 

12. 

0.0006. 

16. 

0.0000002 

5.     0.12|. 

9.     0.6J. 

13. 

0.16f. 

17. 

0.0081. 

6.     0.37f 

10.     0.311 

14. 

0.2^. 

18. 

0.944f. 

Oral 

E 

X  E  R  C 1 S  ES. 

. 

248.  1.   How  many  tenths  in  1^? 

Analysis. — Since  there  are  10  tenths  in  1,  in  1  half  there  is  1  haL  of 
10  tenths,  or  5  tenths,  Ans. 

2.  How  many  hundredths  in  |^  ?     How  many  thousandths? 

3.  How  many  decimal  places  are  required  to  express  tenths  ? 
To  express  hundredths  ?     Thousandths  ?     (Art.  239.) 

4.  In  \  how  many  tenths  ?     In  f  ?    In  f  ? 

5.  In  I  how  many  hundredths  ?     In  |  ? 

6.  How  many  hundredths  in  -^  ?     In  -^  ?     In  ^""^  ? 

7.  How  many  thousandths  in -^  ?     In^\?     In -^^  ? 

Written     Exercises. 

249.  To  Reduce  Common  Fractions  to  Decimals. 

1.   Reduce  f  to  a  decimal  fraction. 

Analysis. — f  of  1  equals  |  of  3.     Since  we  can-  operation. 

not  divide  3  by  8,  we  reduce  it  to  tenths  by  annex-         8  )  3.000 
ing  a  cipher.     (Art.  244,  o°.)     Now  i  of  30  tenths  ^^r       a 

is  3  tenths  and  6  tenths  over.     6  tenths  =  60  hun-  '        ' 

dredths,  and  i  of  GO  hundredths  =  7  hundredths  and  4  hundredths  over. 
But  4  hundredths  =  40  thousandths,  and  i  of  40  thousandths  =  5  thou- 
sandths.    Therefore  f  —  .375.     Hence,  the 

Rule. — Annejc  ciphers  to  the  numerator  and  divide  hy 
the  denominator. 

From  the  right  of  the  quotient  point  off  as  many  decU 
mal  figures  as  there  are  ciphers  annexed. 


Reduction, 


117 


Notes. — 1.  If  tlie  number  of  figures  in  tlie  quotient  is  less  than  the 
number  of  ciphers  annexed  to  the  numerator,  supply  the  deficiency  by 
jprejixiug  ciphem. 

2.  When  the  division  has  been  carried  as  far  as  desirable,  the  remain- 
der may  be  written  over  the  divisor  and  annt.^xed  to  the  quotient. 

8.  If  the  remainder  is  |  or  more,  the  last  decimal  figure  may  be  increased 
by  1.    If  the  remainder  is  less  than  i  the  divisor,  it  may  be  omitted  and  the 


sign  +  annexed  to  the  result. 


Reduce  the  following  fractions  to  decimals 


2. 
3. 
4. 
5. 


i 

4* 
3. 
4* 

5. 
8* 
4 


6. 
7. 
8. 
9. 


10. 
11. 
12. 
13. 


A. 

20' 


20- 
1.3 

"To- 


14. 
15. 
16. 
17. 


Reduce  the  following  mixed  numbers  to  decimals : 

75|.         20.     39|. 
21.     654. 


18. 
19. 


80}. 


22.  8.07^-. 

23.  0.8^V 


24. 
25. 


Reduce  the  following  to  hye  decimal  places : 
26.      f.  27.     |.  28.      f  29.      ^. 


30. 


3  of     Tf 

Vo  of  A- 

■3-  of  i 

4  ^^    8- 

4    nf     6  0 


27.811 
93.18|. 


41 
30  0' 


31.  Reduce  ^  to  the  form  of  a  decimal. 

Analysis — Annexing  ciphers  to  the  numerator 
and  dividing  by  the  denominator,  the  quotient  is  3 
continually  repeated  and  the  remainder  is  always  1. 
Therefore  \  cannot  be  exactly  expressed  by  decimals. 


OPERATION. 

3 )  l.OOQQ 

.3333  etc. 


32. 


Reduce  -^  to  the  form  of  a  decimal. 


Analysis. — The  first  three  quotient  figures 
are  135,  and  the  remainder  is  5,  the  same  as  the 
numerator ;  the  second  three  are  135,  the  same 
set  of  figures  as  before,  and  so  on. 


OPERATION. 


37  )  5.000000 

.135135  etc. 


250.  When  the  numerator,  with  ciphers  annexed,  is  exactly 
divisible  by  the  denominator,  the  decimal  is  called  a  Ter- 
minate decimal. 


251.  When  it  is  not  exactly  cli  vis  idle,  and  the  same  figure  or 
set  of  figures  continually  recurs  in  the  quotient,  the  decimal  is 
called  an  Interminate  or  Circulating  decimal. 

(For  Circulating  Decimals,  see  Art.  877,  Appendix.) 


118  Decimals. 


Addition    of    Decimals. 

252.  Since  decimals  increase  and  decrease  regularly  bj  the 
scale  of  ten.y  it  is  plain  they  may  be  treated  like  integers, 
(Arts.  60,  72.) 

Oral     Exercises. 

253.  1.  "What  is  the  sum  of  A  and  .5  ? 

Analysis. —  .4  =  j\,  and  .5  =  ^q.  Now  4  tenths  and  5  tenths  are  y%, 
or  .9,  Ans. 

2.  What  is  the  sum  of  .04  and  .12  ?     Of  .09  and  .15  ? 

3.  Find  the  sum  of  .006  and  .007.     Of  .008  and.012. 

4.  Find  the  sum  of  ^.07  and  $.05.  How  many  cents  in  $.12 
and  1.18  ? 

5.  Find  the  sum  of  i.60  and  1.40.     How  many  dollars? 

6.  Find  the  sum  of  $.004  and  $.006.     How  many  cents? 

7.  How  many  dollars  in  80  cts.  and  90  cts.? 

Written     Exercises. 

254.  For  Adding  Decimals,  see  Art.  60. 

1.  What  is  the  sum  of  236.0503,  .63,  25.432,  and  345.6414  ? 
Ans.  607.7537. 

Note. — Placing  tenths  under  tenths,  hundredths  nnder  hundredtlis, 
etc.,  reduces  the  decimals  to  a  common  denominator  ;  hence  the  ciphers  on 
the  right  may  be  omitted.     (Arts.  244,  3\) 

2.  What  is  the  sum  of  $53.07 +  $7.923 +  $61.033 +  $60,705  ? 

3.  Find  the  sum  of  15.063+8.0023  +  2.05  +  213.306. 

4.  Find  the  sum  of  40.103  +  217.054  +  385.0063  +  430.00057. 

5.  What  is  the  sum  of  48.05  +  125.006  +  7.0364  +  206.42? 

6.  What  is  the  sum  of  2.0707  +  100.04  +  24.084  +  7.034? 

7.  A  man  bought  a  horse  for  $375.50,  which  was  $35,625 
less  than  what  he  sold  it  for ;  what  did  he  get  for  it  ? 

8.  A  lady  paid  $65,375  for  a  dress,  $375.50  for  a  shawl,  and 
$287,125  for  a  set  of  furs  ;  what  was  the  price  of  all?         ^ 


Suhiraction.  119 

Subtraction    of    Decimals. 

Oral     Exercises. 

255.  1.  From  .9  subtract  .4. 

Analysis. —  .9  -—  f*o,  and  .4  =  j\.  Now  4  tenths  taken  from  9  tenth 
leaves  j%,  or  .5,  Ans. 

2.  From  .23  take  .12.     From  .32  subtract  .24. 

3.  From  .25  take  .08.     From  .42  take  .12. 

4.  What  is  the  dilference  between  f  and  ^^  ? 

5.  What  is  the  difference  between  ^  and  .2  ? 

6.  What  is  the  difference  between  $.50  and  1.25  ? 

7.  What  is  the  difference  between  $.50  and  $.75  ? 

Written     Exercises. 

256.  For  Subtracting  Decimals,  see  Arts.  71,  72. 

1.  From  24.35  subtract  6.2875.     Ans.  18.0625. 

Note. — Writing  the  same  orders  in  the  same  column,  in  effect  reduces 
the  given  numbers  to  a  common  denominator. 

2.  From  $372.06  take  $168,234.     Ans.   $203,826. 

3.  A  lad  bought  a  bicycle  for  $15.37-|-  and  sold  it  for  $12.75  ; 
how  much  did  he  lose  ? 

4.  A  real  estate  dealer  bought  a  house  for  $8256.75,  and 
sold  it  for  $10000  ;  how  much  did  he  make  ? 

5.  A  farmer  gave  20  sheep  worth  $65.50,  3  cows  worth  $100, 
and  2  tons  of  hay  worth  $28.75,  for  a  horse,  and  afterwards 
sold  the  horse  for  $200  ;  how  much  did  he  make  or  lose  by 
his  trades  ? 

6.  What  is  the  value  of  $5263.5— $4236.40  +  1278.80  ? 

7.  What  is  the  value  of  $375.40 +  $478,375  —  1683.10  ? 

8.  What  is  the  value  of  $756.25  — ($175 +  $30 +  $28.60)  ? 

9.  Two  ships  start  from  the  same  island  ;  one  sails  due  north 
461.25  miles,  the  other  due  south  345.16  miles  ;  how  far  apart 
are  they,  and  how  much  farther  has  one  sailed  than  the  other  ? 


120  Decimals. 


Multiplication  of  Decimals. 

Oral     Exercises. 

257.  1.  What  is  the  product  of  3  times  .2  ? 
Analysis. —  .2  =  j^,  and  3  times  y%  are  ^-^  =  .6,  Afis. 

2.  What  is  4  times  .2  ?     3  times  .3  ?     6  times  .4 ? 

3.  How  many  decimal  figures  in  the  product  of  tenths  hy 
units  ? 

4.  If  a  pound  of  tea  costs  $.5  what  will  3  pounds  cost  ? 

5.  What  is  4  times  .03  ? 

Analysis.— .03  —  y|o,  and  4  times  yfo  ^^^  jo%  =  '^^>  ^^^*- 

6.  What  is  5  times  .07  ?     7  times  .06  ?     6  times  .08  ? 

7.  How  many  decimal  figures  in  the  product  of  hundredths 
by  units  ? 

8.  What  will  5  inkstands  cost  at  1.08  apiece? 

9.  What  is  .2  times  .3  ? 

Analysis.— .3  =  -^^  and  .3  =  j\.     Now  y%  x  x%=xf^,  and  yf^  expressed 
decimally  is  ,06,  Ans. 

10.  What  is  .6  multiplied  by  .4?     .5  by  .7  ?     .8  by  .6  ? 

11.  How  many  decimal  figures  in  the  product  of  tenths  by 
tenths  ? 

12.  Multiply  .07  by  A,  using  the  slate  if  necessary. 

Analysis.—  .07  =  y^^  and  .4  -  y\.     Now  y^^  x  y^^  =  yff  „,  which  ex- 
pressed decimally  is  .028,  Ans. 

13.  How  many  decimal  figures  in  the  product  of  hundredths 
by  tenths  ? 

14.  At  $.06  a  pound,  what  will  .5  pound  of  sal  soda  cost? 

15.  Multiply  8  thousandths  by  5  tenths  ?     6  thousandths  by 
7  hundredths  ? 

258.  From  these  examples  we  derive  the  following 

Prixciple. — The  product  of  any  two  decimals  has  as  many 
decimal  figures,  as  both  factors. 


Multiplication.  121 

Written     Exercises. 

259.  To  Multiply  Decimals. 

1.  Multiply  .29  by  .7. 

Analysis. — Multiplying  the  decimals  as  integers  the  .29 

product  is  203 ;  but  as  the  multiplicand  has  two  deci-  t^ 

mal  figures  and  the  multiplier  one,  the  product  must 

have  three.     (Art.  258.)     Hence,  the  .203,   Ans. 

Rule. — Multiply  the  niinibers  as  integers,  and  from 
the  right  of  the  produet  point  off  as  many  figures  for 
decimals  as  there  are  decimal  places  in  both  factors. 

Note. — If  the  product  has  not  as  many  figures  as  there  are  decimals  in 
both  factors,  supply  the  deficiency  by  prefixing  ciphers. 

Ans.  .092024. 
28.3  by  mx4i. 
23.504^2.0006. 
24561  by  .00007. 
62|by  5|xl0.5. 
87Jby2}x.075. 
.000781  by  2.40002. 
278.5by3.87ix2f 

17.  What  cost  465  pounds  of  coffee,  at  31|-  cts.  a  pound  ? 

18.  Find  the  cost  of  608  pounds  of  tea,  at  87-|-  cts.  a  pound. 

19.  Find  the  cost  of  563.5  tons  of  hay,  at  $6.75  a  ton. 

260.  When  the  Multiplier  is  10,  100,  1000,  etc. 

20.  Multiply  4.506  by  100. 

Analysis. — Moving  a  figure  one  place  to  the  operation. 

left  multiplies  its  value  by  10  (Art.  34,  3°),  hence,  4.506 

ijioving  the  decimal  point  one  place  to  the  right  iqq 

multiplies  the  number  by  10,  moving  it  two  places,  

multiplies  the  number  by  100,  etc.     Hence,  the  A71S.    4oO,dOO 

EuLE. — Move  the  decimal  point  in  the  multiplicand  as 
■many  places  to  the  right  as  there  are  ciphers  in  the 
multiplier.     (Art.  91.) 

21.  Mult.  46.3842  by  1000.  23.   Mult.  25.46  by  1000. 

22.  Mult.  6.42302  by  1000,  24.   Mult.  3.004  by  1000. 

6 


2. 

Multiply  23.006  by  .004. 

3. 

5.034  by  .027. 

10. 

4. 

4.0304  by  4.005. 

11. 

5. 

6.4203  by  4.28. 

12. 

6. 

63.0048  by  7.003. 

13. 

7. 

2351  by  3.45. 

14. 

8. 

75}  by' 68. 

15. 

9. 

37iby.7x6i. 

16. 

122  Decimals, 

Division    of    Decimals. 

D  ETJEJLO  P  MENT     OF     I*  R  I N  C  I P  JL  E  S  , 

261.  1.  What  is  the  quotient  of  .8  divided  by  .2  ? 

Analysis.— .8  =  {^  and  .3  =  -f^.     Now  -i^-^-fj^  =  4,  Ans. 

2.  What  is  the  quotient  of  .06  divided  by  .03  ? 

3.  When  tenths  are  divided  by  tenths,  and  hundredths  by 
hundredths,  what  is  the  quotient  ?     Why  ? 

Ans.  Because  they  have  common  denominators.  (Art.  224,  2°.) 

4.  The  product  of  two  numbers  is  3.2  and  one  of  the  factors 
is  4: ;  how  do  you  find  the  other  factor  ?     (Art.  114.) 

5.  Of  what  is  division  the  reverse?     (Art.  101.) 

6.  What  corresponds  to  the  product?     To  the  factors? 

7.  What  is  the  product  of  .06  by  .4  ?  How  many  decimal 
figures  has  it  ? 

8.  If  .024  is  divided  by  .4,  what  is  the  quotient  ?  How  many 
decimal  figures  has  it  ?     Why  ? 

Ans.  Since  division  is  the  reverse  of  multiplication,  the 
dividend  must  have  as  many  decimal  figures  as  the  divisor  and 
qui^tient. 

9.  How  many  decimal  figures  are  there  in  the  product  of  any 
two  factors  ? 

10.  When  tenths  are  divided  by  imits,  how  many  decimal 
figures  in  the  quotient?  Hundredths  hj  units?  Bj  te7ithsf 
Thousandths  by  hundredths  ? 

11.  If  the  dividend  has  five  decimal  places,  and  the  divisor 
three,  how  many  decimal  places  will  there  be  in  the  quotient? 

262.  From  these  examples  we  derive  the  following 

Principles. 

1°.  When  the  decimal  places  in  the  divisor  and  dividend  are 
equal,  the  quotient  is  a  ivhole  number. 

2°.  The  number  of  decimal  places  iti  the  divisor  and  quotient 
must  equal  those  in  the  divide }id. 


Division.  123 

Written     Exercises. 
263.   To  Divide  Decimals. 

1.  What  is  the  quotient  of  .98  divided  by  .7  ? 

Solution. — We   divide   as  in  whole   nmiibers,  and  operation. 

point  off  as  many  figures  for  decimals  in  the  quotient  ^             .7).  98 

as  the  decimal  places  in  the  dividend  exceed  those  in  .       "~1~1 
the  divisor,  wliich  is  one.    Hence,  the 

EuLE. — Divide  as  in  whole  numbers,  and  from  the 
rigJit  of  the  quotient  point  off  as  many  figures  for 
decimals,  as  the  decimal  -places  in  the  dividend  exceed 
those  in  the  divisor. 

If  the  quotient  does  not  contain  figures  enough,  supply 
the  deficiency  hy  prefixing  ciphers. 

Notes. — 1.  When  there  are  more  decimals  in  the  divisor  than  in  the 
dividend,  make  them  equal  by  annexing  ciphers  to  the  latter  before 
dividing.     (Ex.  3.) 

3.  After  all  the  figures  of  th«  dividend  are  divided,  if  there  is  a  remain- 
der, ciphers  may  be  annexed  to  it  as  decimals,  and  the  division  continued 
at  pleasure. 

3.  For  ordinary  purposes,  it  will  be  sufficiently  exact  to  carry  the  quo- 
tient to  four  or  five  places  of  decimals ;  but  when  great  accuracy  is  required, 
it  must  be  carried  farther. 

When  there  is  a  remainder  at  the  close  of  the  operation,  the  sign  + 
should  be  annexed  to  the  quotient  to  show  that  it  is  not  complete. 

2.  Divide  177.6  by  2.4.  Aris.  74. 

3.  Divide  428.1  by  .346.  Ans.  1237.28-f . 

Perform  the  following  divisions  . 

4.  3.560-^3.9.  9.  .00634-^62.  14.  4356.2-^.436. 

5.  4.234-^4.5.  10.  283.25-^82.  15.   643.003^.0:2. 

6.  .04634^5.2.        11.  6432.42-^-7.6.  16.  4873.02^.0064. 

7.  .072-^.8.  12.  .280-^2.4.  17.  756.4^10J. 

8.  1.25^.12J.  13.   0.063-^.09.  18.  1268.2-f-lOf 

19.  If  1.7  yd.  of  cloth  will  make  a  coat,  how  many  coats  will 
81.6  yds.  make  ? 

20.  How  much  broadcloth,  at  $5,675  a  yard.,  can  be  bought 
for  $45.40  ? 


124  Decimals, 

21.  If  a  stage  goes  8.25  miles  an  hour,  how  long  will  it  take 
to  go  125  miles? 

22.  If  a  barrel  of  beef  is  worth  114.25,  how  many  barrels  can 
be  bought  for  $798? 

23.  If  a  steamer  goes  215.6  miles  per  day,  how  long  will  it 
take  to  go  1000  miles  ? 

264.  When  the  Divisor  is  10,  100,  1000,  etc. 

24.  Divide  324.56  by  100. 

Analysis. — As  each  removal  of  a  figure  one  place  opekation. 

to  the  right  diminishes  its  value  ten  times,  moving  a  100  )  324.56 

decimal  point  one  place  to  the  left  divides  the  num-  .  Q  ^A.KiK 

ber  by  ten,  two  places  to  the  left  divides  it  by  100,  etc. 
Hence,  the 

EuLE. — Remove  the  decimal  point  in  the  dividend  as 
many  places  to  the  left,  as  there  are  ciphers  in  the 
divisor.     (Art.  34,  ^°.) 

Perform  the  following  divisions  as  indicated  : 

25.  24.25-^100.  28.     0.08534 -^  1000. 

26.  456.31-^1000.  29.     64.2564-^10000. 

27.  32.463  -^  1000.  30.     56345.27  -^  100000. 

Questions. 

232.  What  is  a  Decimal  Fraction?  Why  called  Decimals?  234.  How 
are  they  expressed?  235.  What  is  the  denominator  of  a  decimal? 
237.  How  determine  the  value  of  a  decimal  figure  ? 

236.  In  writing  and  reading  decimals,  what  should  be  made  the  starting 
point?    240.  How  write  decimals  ?     How  read  them? 

244.  What  is  the  effect  of  removing  the  decimal  point  one  place  to  the 
left?  One  place  to  the  right?  244.  What  is  the  effect  of  annexing  a 
cipher  to  a  decimal  or  removing  one  from  its  right? 

245.  HoAv  reduce  decimals  to  a  common  denominator?  247.  How 
reduce  decimals  to  common  fractions  ?  249.  (Common  fractions  to  decimals  ? 
254.  How  are  decimals  added?    256.  How  subtracted  ? 

259.  How  are  decimals  multiplied?  How  point  off  the  product? 
263.  How  divided?     How  point  off tV- 3  quotient. 


^ 


ECIMAL     CUEEENCY. 


"=^ 


NICKEL 


BRONZE 


United     States     Coins. 

265.  Coins  are  pieces  of  metal  stamped  at  the  Mint,  author- 
ized  by  Government  to  be  used  as  money  at  fixed  values. 

266.  Money  is  the  measure  of  value. 

267.  Currency  is  the  money  employed  in  trade.     It  consists 
of  coins,  bank  bills,  bonds,  bills  of  exchange,  etc. 

268.  A  Decimal  Currency  is  one  whose  orders  or  denomina- 
tions increase  and  decrease  by  the  scale  of  tens. 


126  Decimals, 


United    States   Money. 

269.  II.  S.  Money  is  the  legal  currency  of  the  United  States. 
Its  denominations  are  Eagles,  Dollars,  Dimes,  Cents,  and 
Mills,  which  increase  and  decrease  by  tens.* 

Table. 

10  mills  are  1  cent,  -  -  d, 

10  cents  '*    1  dime,-  -  d. 

10  dimes,  or  100  cts.  ''    1  dollar,  -  dol,  or  I. 

10  dollars  ''    1  eagle,-  -  E. 

270.  The  U.  S.  coins  are  gold,  silver,  nickel,  and  bronze. 

271.  The  Gold  coins  are  the  double  eagle,  eagle,  half  eagle, 
quarter  eagle,  three-dollar  piece,  and  dollar. 

Notes — 1.  The  gold  dollar  is  the  unit  of  'oalue.  Its  standard  weight 
is  35.8  gr.  Troy. 

272.  The  Silver  coins  are  the  dollar,  half  dollar,  quarter 
dollar,  and  dime. 

2.  The  weight  of  tlie  silver  dollar  is  413i  grains.  The  standard  purity 
of  gold  and  silver  coins  is  nine-tenths  pure  metal  and  one-tenth  alloy. 

273.  The  Nickel  coins  are  the  5-cent  and  3-ceiit  pieces. 

274.  The  Bronze  coin  is  the  1-cent  piece. 

275.  The  Dollar  is  the  Unit ;  hence,  dollars  are  written  as 
integers  with  the  sig7i  (-1>)  prefixed  to  them,  and  the  decimal 
point  placed  after  them. 

Cents  occupy  hundredths  place  on  the  right,  and  mills  the 
place  of  thousandths. 

Notes. — 1.  Eao^les  and  dimes  are  seldom  used  in  business  calculations  j 
the  former  are  re?..d  as  dollars,  the  latter  as  cents.  Thus,  15  eagles  are 
read  as  $150,  and  6  dimes  as  60  cents. 

*  The  United  States  adopted  the  decimal  syptem  of  currency  in  1789.  Since  then  it 
has  been  adopted  by  France.  Beliiium,  Brazil,  Bolivia.  Canada,  Chili,  Denmark,  Ecuador, 
Greece,  Germany,  Italy,  Japan,  Mexico,  Norway,  Pera,  Portugal,  Spain,  Sweden,  Swit- 
zerland, Turkey,  U.  S.  of  Colombia,  and  Venezuela. 


TI,  S.  Money,  127 

2.  Cents  occupy  two  places,  lience  if  the  number  to  be  expressed  is  less 
than  10,  a  di^lier  must  be  prefixed  to  the  figure  denoting  them. 

3.  In  business  calculations,  if  the  mills  in  the  result  are  5  or  more,  thej 
are  considered  a  cent ;  if  less  than  5,  they  are  omitted. 

276.  To  reduce  dollars  to  cents,  multiply  them  by  100. 
To  reduce  dollars  to  mills,  multiply  them  by  1000. 

To  reduce  cents  to  mills,  multiply  them  by  10. 

277.  To  reduce  cents  to  dollars  divide  them  by  100. 
To  reduce  mills  to  dollars  divide  them  by  1000. 

To  reduce  mills  to  cents,  divide  them  by  10. 

278.  Dollars,  cents,  and  mills  correspond  to  the  orders  of 
integers  and  decimals,  and  are  expressed  in  the  same  manner. 

Thus,  78  dollars  47  cents  5  mills  are  vmtten,  $78,475. 

Write  the  following  in  like  manner: 

1.  50  dols.  10  cts.  5  mills.  4.     372-3^  dollars. 

2.  •  75  dols.  5  cts.  8  mills.  5.     407  dols.  Vl\  cts. 

3.  627  cents  5  mills.  6.     5260^%  dols. 


Oral     Exe  r  cises. 

1.  Change  5  cents  to  mills.  7.  Eeduce  $4  to  mills. 

2.  Change  8  cents  to  mills.  8.  Keduce  16.10  to  mills. 

3.  Change  40  mills  to  cents.  9.  Reduce  600  cents  to  dollars. 

4.  Change  65  mills  to  cents.  10.  Eeduce  7000  mills  to  dollars. 

5.  Change  83  to  cents.  ii.  Reduce  460  cents  to  dollars. 

6.  Change  $5.20  to  cents.  12.  Reduce  5420  mills  to  dollars. 

13.  A  lad  bought  a  History  for  $1.10,  and  gave  a  two-dollar 
bill  in  payment;  what  change  did  he  receive  ? 

14.  A  dealer  paid  $4.30  for  a  pair  of  boots,  and  sold  them 
for  $5.25  ;  how  much  did  he  make  ? 

15.  If  a  laborer  earns  $1.25  in  one  day,  how  much  can  he 
earn  in  4  days  ? 

16.  What  cost  5  barrels  of  flour,  at  $6.50  a  barrel  ? 

17.  If  5  caps  cost  $3.25,  what  will  1  cap  cost  ? 

18.  At  20  cents  apiece,  how  many  citrons  can  be  bought 
for  $2.40  ? 


128  Decimals, 


Written     Exercises. 

279.  United  States  Money  is  added,  subtracted,  multi- 
plied, and  divided  in  all  respects  like  Decimal  Fractions. 
(Arts.  252-261.) 

1.  A  man  bought  a  cow  for  115.75,  a  calf  for  $2,375,  a  sheep 
for  13.875,  and  a  load  of  hay  for  18.68 ;  how  much  did  he  pay 
for  all  ? 

2.  A  farmer  sold  a  firkin  of  butter  for  19.28,  a  cheese  for 
$1.17,  a  quarter  of  veal  for  $.56,  and  a  bushel  of  wheat  for 
$1.12  ;  how  much  did  he  receive  for  the  whole  ? 

3.  A  man  bought  a  hat  for  $5,375,  a  cloak  for  $35.68,  and  a 
pair  of  boots  for  $4. 75  ;  how  much  did  he  pay  for  all  ? 

4.  What  is  the  sum  of  63  dols.  and  4  cts.,  86  dols.  and  10 
cts.,  and  47  dols.  and  37  cts.  ? 

5.  If  I  pay  $217  for  a  horse  and  $145.50  for  a  buggy ;  what 
is  the  cost  of  both  ?     What  is  the  difference  in  cost  ? 

6.  What  is  the  difference  between  $137.25  +  $65.07  and 
$126,121-  +  $93.06? 

7.  A  man  paid  $63.87^  for  a  sleigh  and  $27.50  for  a  robe, 
and  sold  them  both  for  $185  ;  how  much  did  he  make  ? 

8..  What  will  145  loads  of  wood  cost,  at  $3.25  a  load  r 

9.  Bought  115  barrels  of  apples,  at  $3  a  barrel,  and  sold 
20  barrels  at  $2.50  and  the  remainder  at  $4.25  a  barrel ;  did  I 
gain  or  lose  by  the  operation  ?     How  much  ? 

10.  A  paid  $15  per  acre  for  his  farm  of  365  acres,  and  B 
paid  $23  per  acre  for  his  farm  of  285  acres ;  required  the  dif- 
ference in  tlie  cost  of  their  farms  ? 

11.  A  farmer  bought  165  sheep  at  $6  a  head,  16  cows  at  134, 
and  27  tons  of  hay  at  $21  a  ton,  and  paid  $500  down  ;  how 
much  did  he  then  owe  for  them  ? 

12.  If  a  man  has  a  salary  of  $1800  a  year,  and  pays  $225 
for  his  board,  and  spends  $175  for  clothes  and  $220  for  inci- 
dentals, how  much  will  he  lay  up  in  a  year  ? 

13.  A  grocer  bought  1365  sacks  of  coffee  at  $20  per  sack  ; 
he  sold  563  sacks  at  $25  and  the  balance  at  $27  a  sack  ;  how 
much  did  he  gain  or  lose  ? 


U,  S.  Money.  129 

14.  A  butcher  bought  116  head  of  cattle  at  $47  a  head,  and 
3  times  as  many  sheep  at  |6  a  head ;  how  much  did  he  pay  for 
his  cattle  and  sheep  ? 

15.  How  many  hats  at  $3.75  apiece  can  you  buy  for  $18.75  ? 

16.  If  a  man  pays  $7.25  a  week  for  board,  how  long  can  he 
board  for  $258. 50? 

17.  A  mason  received  $194,375  for  doing  a  job,  which  took 
liim  75|^  days  ;  how  much  did  he  receive  per  day  ? 

18.  At  $1.12|- per  bushel,  how  many  bushels  of  wheat  can 
be  bought  for  $523.75  ? 

19.  If  $1285.25  were  divided  equally  among  125  men,  what 
would  each  receive  ? 

20.  The  salary  of  the  President  of  the  United  States  is 
$50000  a  year  ;  how  much  does  he  receive  per  day  ? 

21.  A  man  paid  $66.51  for  broadcloth,  which  was  $7.39  per 
yard  ;  how  many  yards  did  he  buy  ? 

22.  If  flour  is  $8. 12 J  per  barrel,  how  many  barrels  can  be 
bought  for  $2047.50  ? 

23.  If  556.25  lbs  of  tobacco  cost  $69,532,  how  much  is  that 
a  pound  ? 

24.  At  $47,184  per  ton,  how  many  tons  of  railroad  iron  can 
be  bought  for  $28310.40  ? 

Smout    Methods. 

280.  An  Aliquot  Part  of  a  number  is  an  exact  divisor  of 
that  number. 

Thus,  2,  2i,  3^,  and  5,  are  aliquot  parts  of  10. 

Aliquot     Parts    of    a    Dollar. 


50  cents  =  $i. 

12^  cents  =  $-|-. 

33J  cents  =  %\. 

10  cents  =  $-1^. 

25  cents  =  $J. 

8|  cents  =  %^. 

20  cents  =  %\. 

6J  cents  =  $yV 

16|  cents  =  $|. 

5  cents  =  $^. 

130  Decimals. 

Oral     Exercises. 

281.  1.  What  part  of  II  is  50  cts.  ?    25  cts.?    20  cts.? 

2.  What  part  of  $1  is  \%\  cts.  ?     10  cts.  ?     8-^  cts.  ?     6}  cts.  ? 

3.  Wiiat  will  27  yds.  of  delaine  cost  at  50  cts.  a  yard  ? 

Analysis. — 50  cents  are  %\ ;  therefore  27  yds.  will  cost  37  times  %\,  or 
%^-,  wMcli  are  equal  to  $13i,  or  $13.50,  Ans. 

4.  At  25  cts.  a  pair,  what  cost  75  pairs  of  mittens  ? 

5.  At  12|-  cts.  each,  what  will  be  the  cost  of  72  slates? 

6.  If  you  pay  20  cts.  a  day  for  car-fare,  what  will  be  your 
fare  for  60  days  ? 

7.  At  33J  cts.  a  bushel,  what  will  be  the  cost  of  31  bushels 
of  apples  ?     Of  36  bushels  ?     Of  45  bushels  ?     Of  63  bushels  ? 

8.  At  16|  cts.  a  pound,  what  cost  30  pounds  of  butter? 

9.  What  cost  64  qts.  of  milk  at  6|-  cts.  a  quart  ?    80  quarts  ? 

10.  How  many  melons,  at  \%\  cts.  each,  can  be  had  for  16  ? 

Analysis. — Since  124  cts.  are  $|,  $6  will  buy  as  many  melons  as  %\  is 
contained  times  in  $6,  or  48  melons,  Ans. 

11.  At  $.50  a  pound,  how  many  pounds  of  tea  can  be  bought 
for  111?     For  1184?     For  125  ?     For  $50  ? 

12.  A  farmer  sold  36  bushels  of  oats  at  1.33^  a  bushel,  and 
took  his  pay  in  raisins  at  12 J  cts.  a  pound ;  how  many  pounds 
of  raisins  did  he  receive  ? 

Written     Exercises. 

282.  Price  is  the  money  value  of  a  unit  of  like  things. 

283.  Cost  is  the  sum  paid  for  a  given  number  of  like  things. 

284.  To  find  the  Cost  of  a  number  of  like  things,  when  the  Price 

of  one  is  an  Aliquot  Part  of  $1. 

1.  What  is  the  cost  of  675  Histories,  at  33^  cts.  each  ? 

Analysis.— At  $1  apiece,  they  would  cost  $675.    But  3  )  675 

the  price  is  only  \  of  $1  each  ;  therefore,  the  cost  is  -^^  of  -         r~ 

$675,  which  is  $225.  J «5.     Hence,  the  Ans.  IZb 

EuLE. — Multiply  the  given  ninriber  of  things  hy  the 
fractional  part  of  ^1  which  expresses  the  price  of  One : 
the  result  is  the  cost.     (Art.  208. ) 


TI.  S.  Money,  131 

2.  At  10.50  a  bushel,  what  cost  876  bu.  of  potatoes? 

3.  At  25  cts.  a  yard,  what  will  1200  yards  of  ribbon  cost  ? 

4.  If  I  pay  20  cts.  a  bu.  for  apples,  what  must  I  pay  for  688  bu.? 

5.  What  cost  898  Spellers,  at  12|-  cts.  each  ? 

6.  At  33-|-  cents  a  pound,  what  cost  750  pounds  of  butter  ? 

7.  What  cost  450  boxes  of  lemons,  at  $1.25  a  box  ? 

Analysis. — At  $1  a  box,  they  would  cost  $450.  4  )  $450  at  $1. 

But  the  price  is  $1  +  $J  ;  therefore,  the  cost  is  $450  $112.50  at  ^K 

+  1  of  $450,  which  is  $562.50,  Ans.  *—  ** 

'  $562.50,  Ans. 

8.  At  $1.33^,  what  cost  796  Geographies? 

9.  K  a  man  saves  $1.1 6f  each  week,  how  much  will  he  save 
in  312  weeks  ? 

10.  A  shoe  dealer  sold  at  wholesale  250  pairs  of  slippers  for 
$1.20  a  pair ;  what  was  the  amount  of  his  bill  ? 

285.  To  find  the  Number  of  Things  when  their  Cost  is  given, 
and  the  Price  of  One  is  an  Aliquot  Part  of  $1. 

11.  How  many  gallons  of  milk,  at  $.33^  a  gallon,  can  be 
bought  for  $175  ? 

Analysis. — Since  $1  will  pay  for  3  gallons,  operation. 

$175  will  pay  for  175  times  3  gallons,  or  525  $.33-J^  =  $J 

gallons.     Or,  at  $i  a  gallon,  $175  will  pay  for  |]^75  x  3  ==  525 

as  many  gallons  as  %\  is  contained  times  in  r\      a-j  7K  _^  <!^x  ^9?; 

$175,  or  525  gallons,  Ans.    Hence,  the  '  *      ^ 

EuLE. — Divide  the  cost  by  the  aliquot  -part  of  $1  which 
is  the  price  of  One. 

12.  How  many  yards  of  flannel,  at  50  cts.  a  yard,  can  you 
buy  for  1850  ? 

13.  A  farmer   sold  his   cheese   at   16f  cts.    a  pound,   and 
received  $75  for  it;  how  many  pounds  did  he  sell? 

14.  How  many  bushels  of  oats,  at  50  cts.  a  bushel,  can  be 
had  for  $975  ? 

15.  How  many  cans  of  baking  powder,  at  25  cts.  each,  can 
be  had  for  $240. 50  ? 

16.  How  many  yards  of  silk,  at  $1^  a  yard,  will  $160  buy? 

17.  How  many  hoes,  at  $1.33-^,  can  be  bought  for  $176  ? 


132  Decimals, 

286.  To  find  the  Cost,  when  the  price  per  100  or  1000  is  given. 

18.  What  is  tlie  cost  of  475  oysters,  at  $1.65  per  100? 

Analysts. — At   $1.65  apiece,  475  oysters  would  $1.65 

cost  $1.65  X  475  =  $783.75.    But  the  price  is  $1.65  per  41*^5 

hundred;   tlierefore,  $783.75  is  100  times  the  true 

cost.     To  correct  this  result,  we  divide  it  by  100,  or  1^^  )  1^83.75 

remove  the  decimal  point  two  places  to  the  left.  AllS.    7.8375 

19.  At  $12.60  a  thousand,  what  will  2845  bricks  cost  ? 

Solution.— Multiplying  the  price  of  1000  by  the  $12.60 

number  of  bricks,  and  dividing  the  product  by  1000,  2845 

the  result  is  $35,847,  the  answer  required.     Hence,  the     i|000  )  35~847 

EuLE. — Multiply  the  price  per  hundred  or  thousand  by 
the  given  number  of  things,  and  divide  the  product  by 
100  or  1000,  as  the  case  may  require.     (Art.  118.) 

Note. — The  letter  C  is  sometimes  put  for  kimdred,  and  M  for  thousand. 

20.  What  will  2842  lb.  of  sugar  cost,  at  $12.50  per  hundred  ? 

21.  At  13  J  per  C,  what  will  21264  pounds  of  flour  come  to  ? 

22.  At  $12.50  a  thousand,  what  will  25260  oranges  cost  ? 

23.  At  $25.50  per  hundred,  what  cost  18564  feet  of  boards  ? 

24.  What  cost  1276  cedar  posts,  at  $8.75  per  C.  ? 

25.  What  cost  12250  envelopes,  at  $3.60  per  M.  ? 

26.  At  $6.50  per  thousand,  what  cost  15460  shingles  ? 

27.  At  $12.25  per  hundred,  what  cost  15240  pineapples? 

28.  At  $8.50  per  M.,  what  cost  22580  bricks? 

287.  To  find  the  Cost,  when  the  price  of  2000  pounds  is  given. 

29.  What  cost  2460  pounds  of  coal  at  $6.50  per  ton  ? 

Analysis.— At  $6.50  a  pound,  2460  pounds  will  opekation. 

cost  $6.50  X  2460  =  $15990.00.    But  the  price  is  $6.50  $6.50 

per  ton  of  2000  pounds ;  therefore,  $15990.00  is  2000  2460 

times  the  true  cost.     To  correct  this  result  we  divide  I^QQO  00 

it  by  2000  ;  or  divide  by  2  and  remove  the  decimal  ^^^^  f  lOcV^UAJU 

point  three  places  to  the  left.     Hence,  the  Ans.    $7.99500 

^TjJSE.— Multiply  the  price  of  1  ton  by  the  given  num- 
ber of  pounds  and  divide  the  product  by  2000. 


TJ.  S,  Moiiey.  133 

30.  At  112.50  per  ton,  what  is  the  value  of  8  loads  of  hay, 
each  weighing  1525  pounds  ?     Ans.  $76.25. 

31.  What  is  the  cost  of  12  sacks  of  wool,  each  weighing 
450  pounds,  at  ^25.30  per  ton  ? 

32.  What  is  the  freight  from  London  to  New  York  on  a 
quantity  of  goods  weighing  8540  pounds,  at  ^4.  GO  per  ton  ? 

33.  What  cost  16250  pounds  of  guano,  at  $80J  per  ton  ? 


Accounts    and    Bills. 

288.  An  Account  is  a  record  of  business  transactions. 

289.  Every  business  transaction  has  tico  parties,  a  huyer 
and  a  seller,  called  a  Debtor  and  a  Creditor. 

290.  K  Debtor  is  a  party  who  oioes  another. 

291.  A  Creditor  is  a  party  to  whom  a  debt  is  due. 

292.  A  Ledger  is  the  principal  Book  of  Accounts  kept  by 
business  men.  To  the  Ledger  is  transferred  for  preservation 
and  reference,  a  brief  statement  of  all  the  items  of  the  Day 
Booh  or  Joitrtial,  where  they  are  fully  recorded. 

293.  The  Debits  or  Debts  are  placed  on  the  left,  marked  Dr., 
the  Credits  or  Payments  on  the  right,  marked  Cr. 

294.  The  Balance  of  an  account  is  the  difference  between 
the  Debit  and  Credit  sides. 

295.  A  Bill  is  a  written  statement  of  goods  sold,  or  services 
rendered,  with  their  prices,  etc. 

296.  An  Invoice  is  a  written  statement  of  items  sent  with 
merchandise. 

Note, — Accounts  and  bills  should  always  state  tlie  names  of  both 
parties,  the  place  and  time  of  each  transaction,  the  name  and  price  of 
each  item,  and  the  entire  cost. 

297.  A  Bill  is  Receipted  when  the  words  '^  Received  Pay- 
ment "  are  written  at  tbe  bottom,  and  it  is  signed  by  the 
creditor,  or  by  some  person  duly  authorized. 


134 


Decimals. 


298.  The  following  abbreviations  are  often  used : 


Acct.  or  %,  Account. 

Amt.,  Amount. 

@,  At. 

Bal.,  Balance. 

Do.,  The  same. 


Inst.,  This  month. 

Mdse.,  Merchandise. 

Net.,  Without  Discount. 

Prox.,  Next  month. 

UJt,  Last  month. 


299.  Copy,  extend  the  items,  and  balance  the  following : 

Boston,  May  25tli,  1881. 
James  Browjstell,  Esq., 

Bought  of  Fairman"  &  Lii^coln". 


@  $3.25 


.121 

.15 

.061 


5  yds.  broadcloth, 
3  yds.  cambric, 

3  doz.  buttons, 

6  skeins  sewing  silk, 

4  yds.  wadding,  @,      .08     -     -    -     - 

Amount, 
Received  Payment, 

Fairmak  &  Lincoln. 


(2-) 


Horace  Foote  &  Co., 


New  York,  Feb.  13tli,  1881. 


To  Geo.  Spencer  &  Co.,  Dr. 


1881. 

Feb. 

10 

For  85  lbs.  Coffee,                 @  25 

cts. 

(C 

12 

''    36  lbs.  Tea,                     @  94 

cts. 

a 

S( 

"    63  gal.  Molasses,             @  37|- 

cts. 

a 

13 

''  125  lbs.  Rice,                    @    8^ 

cts. 

(( 

a 

"    75  boxes  Oswego  Stch.,  @  87|- 

cts. 

a 

a 

''     56  lbs.  Bar-soap,             @     6^ 

Amount,   - 

cts. 

Received  Payment, 

Geo.  Spencer  &  Co. 


Accounts  and  Bills. 

(3.) 


135 


Chicago,  May  15th,  1881. 
Messrs.  J.  C.  Geiggs  &  Co., 

Bought  of  Clark  &  M^^Ti^ARD. 


1881, 

May 

1 

150  Spellers,                        @          Q^  cts.  | 

a 

110  Geographies,                 @  S1.20 

2 

72  Roman  Histories,         @  $1.15 

a 

96  Grammars,                    @        65    cts. 

4 

48  Philosophies,                @        56    cts.i 

8 

75  Astronomies,                 @        63    cts. 

Amount,   -    - 

Received  Payment  hy  draft  on  Bosto7i, 

For  Clark  &  Maynard, 

J.  S.  Makn. 

(4.) 

San  Francisco,  Oct.  3,  1881. 

Hexrt  Standart  &  Brother, 

Li  Acct.  iDitli  G.  Atwater  &  Co.,  Dr. 


1881. 

June 

4 

a 

15 

July 

8 

Aug. 

10 

Sept. 

20 

July 

1 

a 

20 

Aug. 

10 

Sept. 

25 

65  tons  E.E.  iron,  @  $45.25 

15  cwt.  Bessemer  steel,  %  $20.50 

18  doz.  Axes,  @  $10.40 

25  Saws,  %    $3.75 

42  cwt.  Lead,  @    $7.40 

Cr, 
500  bbls.  Flour,  @     $5.40 

356  bu.  Wheat,  @    $1.17 

Df t.  on  New  York, 

12  shares  Mining  Stock,        @  $70.00 

Bal.  due,   -    - 
Received  Payment, 

G.  Atwater  &  Co., 


300 


Per  Charles  King. 


136  Decimals, 

Put  the  following  items  into  the  form  of  bills  and  find  the 
amount  of  each : 

5.  Bought  35  cloz.  gloves,  at  14.50  per  doz. ;  95  yds.  hiack 
silk,  at  $.87^  per  yard;  115  yds.  colored  ditto,  at  1.78;  36 
crape  shawls,  at  $32.50  apiece;  65  Broche  ditto,  at  $17.83; 
what  was  the  amount  of  the  bill  ? 

6.  Bought  85  ploughs,  at  19.63  ;  125  hoes,  at  63  cents  ;  94 
shovels,  at  84  cents ;  56  rakes,  at  28  cents  ;  67  axes,  at  $1J3  ; 
what  was  the  amount  of  the  bill  ? 

7.  Bought  96  pair  black  silk  hose,  at  83  cents  ;  85  ditto 
white,  at  87^  cents  ;  135  ditto  worsted,  at  56^  cents  ;  87 
pair  men's  gloves,  at  67  cents  ;  120  pair  ladies'  ditto,  at 
58  cents  ;  75  cravats,  at  96  cents  ;  what  was  the  amount? 

8.  Bought  67  Latin  Headers,  at  63  cents  ;  60  Greek  Eead- 
ers,  at  $1.09  ;  84  Greek  Grammars,  at  68  cents  ;  95  Latin  ditto, 
at  621  cents;  35  Virgil,  at  12.13;  45  Sallust,  at  78  cents; 
52  Cicero's  Orations,  at  75  cents;  what  was  the  amount  of 
the  bill  ? 

9.  Bought  36  pair  of  boots,  at  $5.17  ;  216  pair  thick  shoes, 
at  %l.o1^;  135  pair  gaiters,  at  $1.38  ;  240  pair  buskins,  at  83 
cents ;  134  pair  slippers,  at  68  cents ;  87  pair  rubbers,  at  $1.13 ; 
what  was  the  amount  of  the  bill  ? 

Questions. 

266.  What  is  money?  267.  What  is  currency?  268.  Decimal  cur- 
rency? 269.  U.  S.  Money?  Repeat  the  table.  265.  WJiat  are  coins? 
271.  Name  the  gold  coins  of  U.  S.  272.  The  silver.  273.  The  nickel. 
274.  The  bronze. 

276.  How  reduce  dollars  to  cents?  To  mills?  Cents  to  mills?  277. 
Cents  to  dollars  ?  Mills  to  dollars  ?  Mills  to  cents  ?  279.  Rules  for  cal- 
culating U.  S.  Money  ? 

280.  What  is  an  aliquot  part  of  a  number?  Name  the  aliquot  parts  of 
a  dollar.  284.  How  find  the  cost,  when  the  price  of  one  is  an  aliquot  part 
of  $1  ?  285.  How  find  the  number  of  things,  when  the  cost  is  given  and 
the  price  of  one  is  an  aliquot  part  of  $1  ?  286.  How  find  the  cost,  when 
the  price  per  100  or  1000  is  given? 

288.  What  is  an  account?  290.  A  debtor?  291.  A  creditor?  294. 
The  balance  of  an  account  ?  295.  What  is  a  bill  ?  297.  How  receipted  ? 
296.  An  invoice  ? 


System. 


Definitions. 

300.  Metric  Weights  and  Measures  are  those  whose  units 
increase  and  decrease  regularly  by  the  Decimal  Scale. 

301.  The  Meter  is  the  Base,  and  from  it  the  Metric  System 
derives  its  name.* 

302.  The  Meter  is  one  ien-milUontli  part  of  the  distance 
from  the  Equator  to  the  Pole,  and  is  equal  to  39.37  inches, 
nearly. 

Note. — The  term  Meter  is  from  the  Greek  metron,  a  measure. 

303.  The  Metric  System  has  three  principal  units,  the 
Meter  (meeter),  Liter  (leeter),  and  Gram.  To  these  are 
added  the  Ar  and  8ter,\  for  square  and  cubic  measure.  Each 
of  these  units  has  its  multiples  and  s^Mivisions. 

304.  The  names  of  the  higher  metric  denominations  are 
formed  by  prefixing  to  the  name  of  the  unit,  the  Greek 
numerals,  Deh'a,  Heh'to,  Kilo,  and  Myr'ia. 

Thus,  from  Dek'a,     10,     we  have  Dek'ameter,     10    meters. 
Hek'to,  100,         "       Hek'tometer,  100 
KiFo,      1000,       "       Kil'ometer,      1000     " 
"     Myr'ia,  10000,     ''       Myr'iameter,   10000  '' 

*  This  gystem  had  its  origin  in  France  near  the  close  of  the  last  centnry.    Its  einr 
plicity  and  comprehensiveness  have  secnred  its  adoption  in  nearly  all  the  countries  oi 
Europe  and  South  America. 

Its  use  was  legalized  in  Great  Britain  in  1864,  and  in  the  United  States  in  1866. 

It  is  adopted  by  the  U.  S.  Coast  Survey,  and  is  extensively  used  in  the  Arts  and 
Sciences,  and  partially  in  the  Mint  and  Post  Office. 

t  The  spelling,  pronunciation,  and  abbreviation  of  metric  terms  in  this  work,  are  the 
same  as  adopted  by  the  American  Metric  Bureau,  Boston,  and  the  Metrological  Soc,  N.  Y. 


138  Decimals. 

305.  The  lower  denominations  are  formed  by  prefixing  to 
the  name  of  the  unit  the  Latin  numerals,  Dec'i,  Cen'ti,  and 
Mil'li. 

Thus,  from  Dec'i,     yV>       we  have  Dec'imeter,     -}^      meter. 
''      Cen'ti,   yio.  "         Cen'timeter,  -^^       '' 

-      Milli,    ^^Vo.        "        MirUmeter,    ^\^     - 

Note. — The  numeral  prefixes  are  the  Key  to  the  whole  system,  and 
should  be  thoroughly  committed  to  memory. 


Measures    of    Length. 

306.  The  principal  unit  of  each  table  is  printed  in  capital 
letters  ;  those  in  common  use  in  full-faced  Eoman. 

Ta  b  l  e. 

10  mil'li-yne'ters  {rmn.)     =  1  cen'ti-me'ter,  -  -  cm, 

10  cen'ti-me'ters  =  1  dec'i-me'ter,    -  -  dm. 

10  dec'i-me'ters  =  1  meter,            -  -  m. 

10  me'ters  =  1  dek'a-me'ter,  -  -  Dm. 

10  dek'a-me'ters  =  1  hek'to-me'ter,  -  -  Hm. 

10  hek'to-me'ters  =  1  kilo-me'ter,     -  -  Km. 

10  kil'o-me'ters  =  1  myr'ia-me'ter,-  -  Mm. 

Notes. — 1.  The  Accent  of  each  unit  and  'prefix  is  on  the  first  syllable, 
and  remains  so  in  the  compound  words. 

2.  Abbreviations  of  the  higher  denominations  begin  with   a  capital, 
those  of  the  lower  begin  with  a  small  letter. 

CoMMoi^  Equivalents. 

1  cen'timeter  =  0.3937  inches. 

1  dec'imeter  =  3.937 

1  me'ter  =  39.37*       '' 

1  kil'ometer  —  0.6214  mile. 


*  EstabliBhed  by  Act  of  Congress  in  1866. 


Metric  System.  139 

OyE  DECIMETER, 


llllllll  1  I  I  ll|,il  I  II  I  I  I  llllll  11  liiiil  II  I  ill  II1I1111  li  inl  1111  111  llliii  I  I  III  il  1  II  I  I  I  II  I  li  II  1  ll  II  I  III  11 


100   MiUinietcrs. 

307.  The  Meter  is  the  Standard  Unit  of  length,  and,  Hke  the 
yard,  is  used  in  measuring  cloths,  laces,  short  distances,  etc.* 

308.  Tlie  Kilometer,  like  the  mile,  is  used  in  measuring 
long  distances. 

309.  The  Centimeter  and  Millimeter  are  used  for  minute 
measurements,  as  the  thickness  of  glass,  paper,  etc. 

Note. — The  compound  words  may  be  abbreviated  by  using  only  the 
prefix  and  the  first  syllable  or  letter  of  tbe  unit  ;  thus,  centimeter,  milli- 
meter, centiliter,  milliliter,  centigram,  decigram,  may  be  called  cen- 
tim,  millim,  centil,  decig,  etc. 

310.  The  ap2?roximafe  length  of  1  meter  is  40  inches ;  of 
1  decimeter,  4  inches ;  of  5  meters,  1  rod ;  of  1  kilometer, 
I  mile. 

Note. — Decimeters,  dekameters,  hektometers,  like  dimes  and  eagles,  are 
seldom  used. 

311.  Since  meters,  centimeters,  and  millimeters,  correspond 
to  dollars,  cents,  and  mills,  it  follows  that  metric  numbers  may 
be  read  like  U.  S.  Money.  Thus,  128.375  is  read,  "28  and 
375  thousandths  dollars,"  or  "28  dollars,  37  cents,  5  mills." 

• 

In  like  manner,  28.375  meters  are  read,  "28  and  375  thou- 
sandths meters,"  or  "  28  meters,  37  centimeters,  5  millimeters. 

312.  Eead  the  following  : 

1.  14.5  m.  5.  47.3  Dm.  9.  89.63  Hm. 

2.  236.4  m.  6.  83.25  Dm.  10.  434.5  Km. 

3.  78.35  m.  7.  568  Hm.  11.  65.48  Km. 

4.  23.7  Dm.  8.  648.8  Hm.  12.  9.237  Km. 

*  It  is  important  for  the  teacher  to  show  the  class  a  meter  stick,  with  its  subdivi- 
sions marked  on  one  side,  and  halves,  quarters,  etc.,  on  the  other. 


140  Decimals. 

313.  To  write  Metric  Numbers  decimally  in  terms  of  a  given  Unit. 

1.  Write  7  Hm.  9  m.  3  dm.  5  cm.  in  terms  of  a  meter. 

Explanation.  —  We  write  meters    in    units  operation. 

place,  on  the  left  of  the  decimal  point,  the  Dm.  in         709.35  m.,    Ans. 
tens  place,  the  Hm.  in  hundreds  place,  etc.,  and  the 

decims.  in  tenths  place,  centims.  in  hundredths,  etc.,  as  we  write  the  orders 
of  integers  and  decimals  in  simple  numbers.     Hence,  the 

Rule. — Write  the  given  unit  and  the  higher  denoini- 
nations  in  their  order,  on  the  left  of  a  decimal  point,  as 
integers,  and  those  helow  the  unit,  on  the  right,  as 
decimals. 

Note. — If  any  intervening  denominations  are  omitted  in  the  given 
number,  their  places  must  be  supplied  by  ciphers. 

Write  the  following  as  meters  and  decimals  : 

2.  256  millimeters.     A7is.  0.256  m. 

3.  8  decimeters  4  centimeters. 

4.  25  meters  3  centimeters. 

5.  348  dekameters  43  centimeters. 

6.  465  hektometers  48  millimeters. 

7.  4725  meters  25  centimeters. 

8.  4  Km.  8  Hm.  6  Dm.  4  dm.  5  cm.  3  mm. 

9.  23  Km.  6  Hm.  8  dm.  6  cm. 

314.  To   reduce   Metric   Numbers  from  higher  denominations  to 

lower,  and  from  lower  to  higher. 

1.  Eeduce  45  meters  to  millimeters. 

45  m 
Solution.— Since  1  m.  =  1000  mm.,  45  meters  ^^    ' 

must  equal  45  x  1000,  or  45000  mm.,  Ans.  -2??_ 

Ans.  45000  mm. 

2.  Eeduce  64000  millimeters  to  meters. 

Solution.— In  1000  mm.  there  is  1  m.,  and  in  64000        1000  )  64000 

mm.  there  are  as  many  meters  as  1000  is  contained  .      ~~r. 

times  in  64000,  or  64  meters.     Hence,  the  ^^^^-   "^  ^• 

EuLE. — Move  the  decimal  point  one  place  to  the  right 
or  left,  as  the  case  may  require,  for  each  denomination 
to  which  the  given  number  is  to  he  reduced. 


Metric  System.  141 

3.  Reduce  25.7  Km.  to  meters.    Ans.  25700  m. 

4.  Eeduce  43.4  m.  to  millimeters. 

5.  Eeduce  65.3  Dm.  to  decimeters. 

6.  Eeduce  84.25  Km.  to  centimeters. 

7.  Eeduce  4823.6  meters  to  Hektometers."^M5.  48.236  Hm. 

8.  Eeduce  36482.9  m.  to  kilometers.     Ans.  36.4829  Km. 

9.  Eeduce  28526  mm.  to  meters  and  decimals. 

10.  Eeduce  48639  cm.  to  meters  and  decimals. 

11.  Eeduce  438.6  m.  to  millimeters. 

12.  Eeduce  738.4  Dm.  to  centimeters. 

Measures    of    Surface. 

315.  A  Surface  is  that  which  has  length  and  breadth  only. 

316.  The  Measuring  Unit  of  Surfaces  is  a  Square,  each  side 
of  which  is  a  Linear  Unit. 

317.  A  Square  is  a  figure  which  has  four  equal  sides  and 
four  equal  angles,  called  right  angles. 

Tabl  e. 

100  sq.  mil'li-me'ters  (sq.  mm.)  =     1  sq.  cen'ti-me'ter,  sq.  cm. 
100  sq.  cen'ti-me'ters  =     1  sq.  dec'i-me'ter,    sq.  dm. 

100  sq.  dec'i-me'ters  =1         '  , ,  '  q-     - 

^  t  or  cent  ar,  ca. 

100  sq.  me'ters  '  =  ]  ^  '\  ^^'^^'^^'^^'  *?•  Dm. 

^  ( or  Ar,  A. 

100  sq.  dek'a-me'ters  =  |  ^  '^^  ['f^'to-'^e'ter, .?.  Hm. 

^  f  or  hek  tar,  Ha. 

100  sq.  hek'to-me'ters  =     1  sq.  kH'o-me'ter,    sq.  Km. 

COMMOIis"   EqUIVALEIs'TS. 

1  sq.  centim.  =  0.1550  sq.  in. 

1  sq.  decim.  r=  0.1076  sq.  ft. 

1  sq.  meter  =  1.196  sq.  yd. 

1  ar  —  3.954  sq.  rods. 

1  hektar  =  2.471  acres. 

1  sq.  kilo  =  0.3861  sq.  mile. 


142  Decimals, 

318.  The  sq.  meter  is  used  in  measuring  ordinary  surfaces, 
as  floors,  ceilings,  etc.  ;  the  ar  and  hektar  in  measuring  land; 
and  the  sq.  kilometer  in  measuring  States  and  Territories. 

Note. — The  term  ar  is  from  the  Latin  araa,  a  surface. 

319.  The  approximate  area  of  a  sq.  meter  is  lOf  sq.  ft.,  or 
\\  sq.  yd.,  and  of  the  hektar  about  %\  acres. 


a 


320.  The  scale  of  surface  measure  is  100  (10  x  10). 
That  is,  100  units  of  a  lower  denomination  make  a 
unit  of  the  next  higher;  hence,  each  denomination 
must  haye  two  places  of  figures.  ^^"  ^®"t^™- 

Thus,  23  Ha.  19  a.  25  ca.,  written  as  ars,  is  2319.25  a.,  and  may  be  read 
"2319  ars  and  25  centars."  If  written  as  hektars,  it  is  23.1925  Ha.,  and 
may  be  read  "  23  hektars  and  1925  centars." 

1.  Write  78.29  a.  as  centars,  also  as  hektars. 

2.  Write  9  sq.  m.  as  sq.  dm.     Write  7  sq.  cm.  as  sq.  mm. 

3.  In  3246  ca.,  how  many  ars  ?    In  63.42  ars,  how  many  Ha.  ? 

Measures    of    Solids. 

321.  A   Solid  is    that    which    has    length,    breadth,   and 

thickness. 

Ta  b  le. 

1000  cu.  mirii-me'ters  {cu,  mm.)  =  1  cu.  cen'ti=me'ter,  cu.  cm. 

1000  cu.  cen'ti-me'ters  =  1  cu.  dec'i-me'ter,  cu.  dm. 

1000  cu.  dec'i-me'ters  =  1  cu.  meter,  cu.  m. 

10  dec'i-sters  =  1  ster,  st 

10  sters  =  1  dek'a-ster.  Dst. 

Common"  Equivalents. 
1  cu.  centimeter        =         0.061  cu.  in. 
1  cu.  decimeter  =         61.022  cu.  in. 

1  cu.  meter  =         1.308  cu.  yds. 

Note.— The  ster  =  .2759  cord  is  seldom  used. 

322.  The  Measuring  Unit  of  solids  is  a  Cube,  the  edge  of 
which  is  a  Linear  Unit. 


Metric  System,  143 

323.  A  Cube  is  a  regular  solid  bounded  by 
six  equal  squares  called  its  faces.  Hence,  its 
length,  breadth,  and  thickness  are  equal. 

A  Cubic  Centimeter  is  a  cube,  each  side  of  cu-  cm. 

which  is  a  square  cejitinieter. 

324.  The  cubic  meter  is  used  in  measuring  ordinary  solids, 
as  timber,  excavations,  embankments,  etc. 

When  applied  to  fire- wood,  it  is  sometimes  called  a  Ster,  atid 

is  equal  to  about  35-|-  cubic  feet. 

Note. — The  ciibic  decimeter  when  used  as  a  unit  of  dry  or  liquid 
measure  is  called  a  Liter. 

325.  The  units  of  cubic  measure  increase  by  the  scale  of 
1000  (10  X  10  X  10) ;  hence,  each  denomination  must  have  three 
places  of  figures. 

1.  Express  6000  cu.  mm.  as  cu.  centimeters. 

2.  Express  8000  cu.  dm.  as  cubic  meters. 

3.  Express  86.005  cu.  dm.  as  cu.  meters ;  as  cii.  cm. 

4.  Write  0.6235  cu.  m.  as  cu.  dm. ;  as  cu.  cm. 

5.  In  862  cu.  dm.,  how  many  cu.  meters  ?  In  250  cu.  m. 
how  many  cubic  decimeters  ? 

Measures    of    Capacity. 

326.  The  Liter  is  the  principal  unit  of  Dry  and  Liquid 
Measure,  and  is  equal  in  volume  to  a  cubic  decimeter. 

Table. 

10  mil'li-li'ters  (ml.)     =  1  cen'ti-li'ter  -  -  -  cl. 

10  cen'ti-h'ters  =  1  dec'i-li'ter    -  -  -  dl 

10  dec'i-li'ters  =  1  liter      -    .  .  .  l, 

10  li'ters  =  1  dek'a-li'ter  -  -  -  Dl 

10  dek'a-li'ters  =  1  hek'to-li'ter  -  -  HI 

10  hek'to-h'ters  =  1  kil'o-li'ter    -  -  -  Kl. 

10  kil'o-li'ters  =  1  myr'ia-li'ter  -  -  Ml. 


144 


Decimals. 


^g 

% 

««^ 

^ 

•^SUm 

^ 

1  cubic  centimeler  =  1  milliliter  of  water. 


CoMMOi^  Equivalents. 


1  liter 

-         61.022  cu.  inches. 

1  liter 

=         1.056"/  liquid  quarts 

1  liter 

:         0.908  dry  quarts. 

1  hektoliter       = 

3.531  cu.  feet. 

1  hektoliter       = 

26.417  gallons. 

1  hektoliter       — 

2.837  bushels. 

327.  The  Centiliter  is  a  little  less 
than  \  gill,  and  is  used  for  measuring 
liquids  in  small  quantities. 

The  Liter  is  used  in  measuring 
milk,  wine,  and  small  fruits,  and  is 
about  equal  to  a  quart. 

The  Hektoliter  is  used  in  measur- 
ing grain  and  liquids  in  casks,  and  is  equal  to  about  26-|-  gal., 
or  2|-  bushels. 

1.  Express  8.53  1.  as  centiliters.     As  deciliters. 

2.  Express  4. 640  kiloliters  as  liters.     As  hektoliters. 

3.  How  many  deciliters  in  8  liters  ?     In  9.35  liters? 

4.  How  many  liters  in  6.358  centiliters  ?     In  800  cl.  ? 

5.  In  8500  liters  how  many  kiloliters  ?    How  many  HI.  ? 


Weight. 

328.  The  Gram  is  the  'principal  unit  of  weight,  and  is  equal 
to  a  cubic  centimeter  of  distilled  water  at  its  greatest  density, 
viz.,  at  4°  Centigrade,  or  39.2°  Fahrenheit. 


Metric  System, 


145 


Ta  b  l  e 


10  mil'li-grams  {mg^     = 

10  cen'ti-grams  = 

10  dec'i-grams  =: 

10  grams  = 

10  dek'a-grams  = 

10  hek'to-grams  = 

10  kiro-grams  = 

100  m}T'ia-grams  == 


1  cen'ti-gram     - 

1  dee'i-grani 

1   CRAM  -       - 

1  dek'a-gram 
1  liek'to-gram    - 
1  kiro-gram 
1  myr'ia-gram    - 
1  tonneau  or  Ton 


eg. 
clg. 

9- 
Dg. 

Hg. 

Kg. 

Mg. 

T. 


IDg. 


1  gram. 


COMMOI^ 


idg. 


1  ds. 


leg 


Ics 


1  gram 


1  kilogram      = 
1  metric  ton   = 


1  gram  = 

1  gram  z=z 

1  kilogram  =: 

1  metric  ton  = 


Equiyalents. 
1  cu.  centim.,  or 
1  millil.  of  water. 
1  cu.  decim.,  or 
1  liter  of  water. 
1  cu.  meter,  or 
1  kiloliter  of  water. 
15.432  grs.  Troy. 
0.03527  oz.  Av. 
2.2046  lbs.  Av. 
1.1023  tons. 


© 

Img. 


329.  The  Gram  is  used  in  weighing  gold,  silver,  jewels,  and 
letters,  and  in  mixing  medicines, 

7 


146  Decimals. 

330.  The  Kilogm7n,  (often  called  hilo)  is  used  in  weighing 
common  articles  ;  as  sugar,  tea,  butter,  etc. 

The  Metric  ton  is  used  in  weighing  heavy  articles ;   as  hay, 
coal,  etc. 

Notes. — 1.  The  kilo  is  equal  to  2 1  lbs.,  nearly ;  the  metric  ton  about 
2300  pounds.* 

2.  The  nickel  5-cent  piece  weighs  5  grams.     The  silver  I  dollar  12i 
grams.     The  silver  dime  weighs  2i  grams.     The  silver  |-  dollar  Q\  grams. 

3.  The  weight  of  a  letter  for  single  postage  must  not  exceed  15  grams, 
or  3  nickels. 

1.  Express  6.354  g.  as  decigrams.     As  centigrams. 

2.  Write  5834  mg.  as  dg.     As  eg.     As  grams. 

3.  How  many  grams  in  78.45  Dg.  ?     How  many  Kg.  ? 

4.  How  many  kilos  in  3.54  T.  ?     How  many  Dg.  ? 

5.  Express  1  g.  in  the  decimal  part  of  a  kilo. 

6.  Express  a  kilo  in  the  decimal  part  of  a  ton. 

7.  Express  2.0005  T.  as  grams. 

331.  To  Add,  Subtract,  Multiply,  and  Divide  IVIetric  numbers. 

A2)ply  the  correspo?idmg  rules  of  decimals  or  U.  S.  money. 
(Art.  279.) 

1.  What  is  the  sum  of  45.68  Dm.,  63.4  Hm.,  and  6845  cm.? 

,,r  ...       1  V.  .  456.8 

Solution. — Writing  the  numbers  as  meters 

and  decimals  of  a  meter,  the  principal  unit  of  ooiu.u 

the  table,  and  adding,  we  have  6865.25  meters.  68. 45 

Ans.  6865.25  m. 

2.  Find  the  sum  of  24.35  m.,  6.425  m.,  32.7  m.,  and  42.26  m. 

3.  What  is  the  difference  between  8.5  kilograms  and  976 
grams  ? 

Solution.— 8.5  kilos  —  .976  kilos  =  7.524  kilos,  Ans. 

4.  From  1  hektoliter  of  oil,  36  liters  were  drawn  out ;  how 
many  liters  remained  ? 

5.  How  much  silk  is  there  in  12J  pieces,  each  containing 
48.75  meters  ? 

Solution.— 48.75  m.  x  12.5  =  609.375  m.,  Ans. 


Metric  System.  147 

6.  It  is  285  meters  around  my  garden  ;  how  many  Km.  shall 
I  walk  in  a  week  by  going  twice  around  it  every  day  ? 

7.  At  16.50  a  meter,  what  will  37  meters  of  silk  cost? 

8.  What  cost  24  meters  of  fringe,  at  12.25  a  meter? 

9.  How  many  cloaks,  each  containing  5.68  meters,  can  be 
made  from  426  meters  of  cloth  ? 

Solution. — 426  m.  -7-  5.68  m.  =  75  cloaks,  Ans. 

10.  If  a  car  goes  160  Km.  in  6  hours,  how  far  does  it  go  in 
1  hour  ? 

11.  How  many  Km.  in  85.72  m.  multiplied  by  2036  ? 

12.  If  the  price  of  1  liter  of  milk  is  6  cents,  what  cost  75  liters  ? 

13.  At  12  cents  a  liter,  what  cost  4.5  liters  blackberries  ? 

14.  If  1  hektoliter  of  wheat  costs  $3.50,  what  will  234  hekto- 
liters  cost  ? 

15.  A  man  paid  !i5281.75  for  245  hektoliters  of  oats;  what  was 
the  price  of  1  hektoliter  ? 

16.  What  cost  46.25  kilos  of  butter,  at  10.50  per  kilo  ? 

17.  At  $1.28  per  kilo,  what  will  82.5  kilos  of  tea  come  to  ? 

18.  At  $16  a  ton,  what  will  the  coal  cost  to  supply  a  factory 
a  week,  if  25  kilos  are  burned  each  day  ? 

19.  If  735  kilos  of  flour  are  distributed  among  35  persons, 
how  many  kilos  'vvill  each  person  receive  ?* 

332.  The  contents  of  Rectangular  Surfaces  are  found  by 
multiplying  the  length  by  the  Ireadtli. 

20.  A  garden  is  18  meters  long  and  12.5  meters  wide  ;  how 
many  square  meters  does  it  contain  ? 

Solution.— The  product  of  18  x  13,5  =  225  sq.  m.,  Ans. 

21.  How  many  sq.  meters  in  a  blackboard  2.5  meters  long 
and  1.2  meters  wide  ? 

22.  If  a  room  is  8.4  meters  long  and  4.5  meters  wide,  how 
many  square  meters  of  carpeting  will  it  take  to  cover  the  floor  ? 

23.  How  many  sq.  meters  of  flagging  in  a  side-walk  35.5 
meters  long,  and  2.4  meters  wide? 

*  For  reducing  Metric  to  common  Weights  and  Measures,  etc.,  see  Ai't.  405, 


148  Decimals 

24.  How  many  centars  in  a  piece  of  land  45  meters  long,  and 
23.2  meters  wide? 

333.  The  contents  of  Rectangular  Solids  are  found  by  multi- 
plying the  length,  breadth,  and  thickness  together. 

25.  How  many  cu.  meters  of  earth  in  a  mound  whose 
length,  breadth,  and  height  are  each  6.4  meters. 

Solution. — 6.4  x  6.4  x  6.4  =  262.144  cu.  meters,  Ans. 

26.  How  many  cu.  meters  of  earth  must  be  removed  in  dig- 
ging a  cellar  23.4  meters  long,  15.2  m.  wide,  and  2.4  m.  deep? 

27.  How  many  loads  of  earth  each  equal  to  a  cu.  meter,  will 
it  take  to  fill  an  excavation  4  dekameters  long,  8  meters  wide, 
and  2. 4  meters  deep  ? 

28.  At  $1.45  a  cu.  meter,  what  will  be  the  cost  of  digging  a 
trench  2  dekameters  long,  2  meters  wide,  and  1.5  meters  deep? 

29.  At  12.50  a  ster,  what  is  the  cost  of  a  pile  of  wood  3 
meters  long,  1.5  m.  wide,  and  1.1  m.  high  ? 

30.  What  is  the  value  of  a  nugget  of  gold  2.6  cm.  long,  2.3  cm. 
wide,  and  0.65  cm.  thick,  at  $15.40  a  cu.  centimeter? 

QU  ESTIO  N  S. 

300.  What  are  Metric  weights  and  measures  ?  301.  What  is  the 
Base?  304.  How  are  the  names  of  the  higher  denominations  formed? 
305.  The  lower  ?    306.  Repeat  the  table  of  measures  of  length. 

307.  What  is  the  standard  unit  of  length  ?  For  what  used  ?  308.  The 
kilometer?    311.  How  read  metric  numbers ?     318.  How  write  them ? 

314.  How  reduce  metric  numbers  from  higher  to  lower  denominations  ? 
From  lower  to  higher?  317.  Repeat  the  table  of  measures  of  surface. 
318.  For  what  is  the  square  meter  used?  The  sq.  kilometer?  The  ar 
and  hektar? 

321.  Repeat  the  measures  of  solids.  324  For  what  is  the  cu.  meter 
used  ?  When  called  a  ster  ?  826.  Repeat  the  table  of  measures  of  capac- 
ity.    327.  For  what  is  the  liter  used?     The  hektoliter  ? 

828.  Repeat  the  table  of  weight.  329.  For  what  is  the  gram  used  ? 
830.  The  kilogram?  The  metric  ton?  381.  How  are  metric  nimibers 
added,  subtracted,  etc.  ? 


^i=-€^ — '■ — 

^))  C)  M  P  O  U  ^'  I) 


U  M  B  E  R  S . 


-\^  i'^'in  >~ 


Definitions. 

334.  A  Simple  Number  is  one  wliicli  expresses  one  or  more 
units  of  the  same  navie  or  (lenoniination  j  as  five,  4  feet,  etc. 

335.  A  Compound  Number  expresses  units  of  two  or  more 
denominations  of  the  same  kind,  which  increase  and  decrease 
by  varying  scales ;  as,  3  yards  2  feet  4  inches.  But  2  feet  and 
4  pounds  is  not  a  compound  number,  for  the  units  are  unlike. 

Note. — Compound  Numbers  are  often  called  Denominate  Numhers. 
The  term  denomination  is  a  name  given  to  the  different  units  of  weights 
and  measures. 

Linear   Measure. 

336.  A  Measure  is  a  standard  unit  established  by  law  or 
custom,  by  which  the  length,  surface,  capacity,  and  weight 
of  things  are  estimated. 

337.  Linear  Measure  is  used  in  measuring  lines  and  dis- 
tances. 

338.  A  Line  is  that  which  has  length  only. 

Ta  b  l  e. 


12 

inches  {in.)          = 

1  foot,     -    ■ 

■    -    fi- 

3 

feet                        =r 

1  yard,     -     - 

■  -  yd- 

5i 

yds.,  or  IGJ  ft.     = 

1  rod,       -     - 

■     -     rd. 

40 

rods                       =: 

1  furlong,     ■ 

■    -    fur 

320 

rods,  or  5280  ft.  = 

1  mile,     -     ■ 

■    -    mi. 

3 

miles                     = 

1  league. 

■    '    I 

339.  The  Standard  Unit  of  length  is  the  Yard,  which  is 
used  in  measuring  cloths,  laces,  ribbons,  etc.    (Art.  900,  App.) 


150 


Compoimd  Numbers, 


Oral     Exer  cises. 

340.     1.  Draw  a  line  4  inches  long.    A  foot.     A  yard. 

2.  How  long  is  this  book  ?    Your  slate  ?    How  wide  ? 

3.  How  long  is  this  table  ?     How  wide  ?     How  high  ? 

4.  In  6  feet  how  many  inches  ?     In  8  ft.  ?     In  9  ft.  ? 

5.  How  many  feet  in  7  yards  ?     In  15  yds.  ?    In  20  yds.  ? 

6.  In  120  inches  how  many  feet  ?     How  many  yards  ? 

7.  How  many  feet  in  4  rods  ?    In  5  rods  ? 


Square    Measure. 

341.  Square   Measure  is   used  in  measuring  surfaces ;  as, 
flooring,  land,  etc. 

342.  A  Surface  is  that  which  has  lengtli  and  hreadth  only. 

343.  An  Angle  is  the  opening  between 
two  lines  which  meet  at  a  point,  as  BAG. 

The  Lines  AB  and  AC  are  called  the 
sides ;  and  the  Point  A,  at  which  they 
meet,  the  Vertex  of  the  angle, 

344.  When  two  straiglit  lines  meet  so 
as  to  make  the  tivo  adjacent  angles  equal, 
the  lines  are  Perpendicular  to  each  other, 
and  the  two  angles  thus  formed  are  called 
Right  Angles ;  as,  ABC,  ABD. 

345.  A  Square  is  a  rectilinear 
figure  which  has  four  equal  sides, 
and  four  right  angles. 

346.  The  yneasuring  unit  of  sur- 
faces is  a  Square,  each  side  of  which 
is  a  linear  unit. 


-.A,, 

ir.' 

g 

1 

1. 

^ 

D 


347.  The  Area  of  a  figure  is  the 
quantity  of  surface  it  contains. 


1    !!'  II  '!| 

il  il  11  |ii|  ill  ii 
hliiiiiiil    l\,l  1 

1  1 

1 

III 

'li  '■mi  |i    ii 
Il   il  1     II 
II  nil  ml  III  II 

i 

1  III  T.    "::    III' 

3  ft.  X  3  ft.  =  1  sq.  yd. 


Cubic  Measure, 


151 


Ta  b  le  . 

144  square  inches  {sq,  in.)  =  1  square  foot,   -  -  sq.  ft. 

9  square  feet  =  1  square  yard,  -  -  sq.  yd. 

30J  sq.  yds.,  or  272^  sq.  ft.  =  1  square  rod,    -  -  sq.  rd. 

160  square  rods  :=  1  acre,     -     -     -  -  A. 

640  acres  =  1  square  mile,  -  -  sq.  ml. 

(For  Surveyor's  Measure,  see  Art.  889,  Appendix.) 

Oral     Exercises. 

348.     1.  How  many  sq.  inches  in  2  sq.  feet? 

2.  In  8  sq.  yds.  how  many  sq.  ft.  ?     In  15  sq.  yds.  ? 

3.  In  2  acres  how  many  sq.  rods  ?     In  3  acres  aud  5  sq.  rd.  ? 

4.  How  many  sq.  ft.  in  288  sq.  inches  ? 

5.  In  320  sq.  rods  how  many  acres  ? 


Cubic    Measure. 

349.  Cubic  Measure  is  used  in 
measuring  solids  or  volume. 

350.  A  Solid  is  that  which  has 
length,  breadth,  and  thickness;  as, 
timber,  boxes  of  goods,  etc. 

351.  A  Cube  is  a  regular  solid 
bounded  by  six  equal  squares  called 
its  faces.  Hence,  its  length,  breadth, 
and  thickness  are  equal  to  each  other. 

352.  The  measuring  unit  of  solids  is  a  Cube  the  edge  of 
which  is  a  linear  unit. 

Ta  b  l  e. 


1728  cubic  inches  (6'?^  in.)   =     1  cubic  foot. 

-     cu.  ft. 

27  cubic  feet                      =     1  cubic  yard,     - 

-     cu.  yd 

128  cubic  feet                     — -     1  cord  of  wood. 

-   a 

152 


Compound  Numhers. 


353.  A  Cord  of  wood  is  a  pile  8  ft.  long,  4  ft.  wide,  and 
4  ft.  high ;  f or  8  x  4  x  4  =  128. 

354.  A  Cord  Foot  is  one  foot  in  length  of  such  a  pile  ; 
hence,  1  cord  foot  =  16  cu.  feet;  8  cord  ft.  ==  1  cord. 

Oral     Exercises. 

355.  1.  How  many  cubic  inches  in  2  cubic  feet  ? 

2.  How  many  cu.  feet  in  2  cubic  yards  ?     In  3  cu.  yards  ? 

3.  How  many  cubic  feet  in  2  cords  ?     In  3  cords  ? 

4.  In  54  cu.  feet,  how  many  cu.  yards  ?     In  126  cubic  feet  ? 

5.  How  many  cord  feet  in  3  cords  of  wood?     In  5  cords  ? 

6.  In  32  cord  feet  how  many  cords  ?    In  72  cord  feet  ? 

Liquid    Measure. 

356.  Liquid    Measure    is    used    in    measuring   millc,    oil^ 
ioine,  etc. 


Ta  b  le 


4    gills  {(ji.) 

==     1  pint,     -     - 

-    pL 

2     pints 

=     1  quart,  -     - 

-     qt 

4     quarts 

^     1  gallon,  -     - 

-    gal 

31|-  gallons 

=     1  barrel,  -     - 

-    lav,  or  libl 

63     gallons 

=     1  hogshead, - 

-    liM. 

pt.    gl. 


357.   The  Standard  Unit  of  Liquid  Measure  is  the  gallon, 
which  contains  231  cubic  inches. 

Note.— The  harrel  and  hogshead,  as  units  of  measure,  are  chiefly  used 
in  estimating  the  contents  of  cisterns,  reservoirs,  etc. 


Dry  Measure. 


153 


Oral     Exercises. 

358.     1.   How  many  quarts  in  20  pints  ?     In  36  pts.  ? 

2.  In  24  pints  how  many  quarts  ?     In  40  pts.  ? 

3.  How  many  gallons  in  2  hogsheads  ?     In  5  hhd.  ? 

4.  How  many  qts.  in  12  gal.  of  milk  ?     In  15  gallons  ? 

5.  What  is  the  cost  of  5  gal.  of  syrup  at  60  cts.  a  gal.  ? 


Dry    Measure. 

359.   Dry    Measure    is    used    in  measuring  grciiyi,  fruit, 
salt,  etc. 

Tab  le. 


2  pints  (pt.)  =     1  quart,  - 

8  quarts  =:     1  peck,    - 

4  pecks,  or  32  qts.  =     1  bushel, 


qt. 
hii. 


360.  The   Standard   Unit  of  Dry   Measure   is  the   huslml, 
which  contains  2150.4  cubic  inches.* 

Note. — The  dry  quart  is  equal  1|  liquid  quart  nearly. 


Oral     Exercises. 

361.     1.  How  many  pints  in  12  quarts  ?     In  25  quarts? 

2.  How  many  pecks  in  40  qts.  of  chestnuts  ? 

3.  How  many  bushels  in  72  pecks  ? 

4.  If  you  pay  40  cts.  for  J  bushel  of  apples,  what  must  you 
pay  for  5  bushels  ? 

5.  If  I  buy  a  bushel  of  walnuts  for  $3,  and  sell  them  at 
5  cts.  a  pt.,  how  much  shall  I  make  ? 

6.  How  many  bu.  in  36  pecks  ?     In  96  quarts  ? 

*  For  the  standard  weight  of  a  hushel  of  different  grains,  see  Art.  896,  Appendix. 


154  Compound  Number's. 

Troy    Weight. 

362.  Troy  Weight  is  used  in  weighing  gold,  silver,  etc. 

Ta  b  le  . 

24  grains  {gr.)        z=     1  pennyweight,  -  -  2^^^' 

20  pennyweights     =     1  ounce,      -     -  •■  -  oz. 

12  ounces  =     1  pound,     -    -  -  -  Ik 


lb.    ;  oz.  pwt.  gr. 

363.  The  Standard  Unit  of  weight  in  the  United  States,  is 
the  Troy  Pound. 

1.  How  many  grains  in  6  pennyweights  ? 

2.  How  many  pwt.  in  6  lb.  7  oz.  ?     In  8  lb.  5  oz.  ? 

3.  How  many  ounces  in  8  pounds  of  silver  ? 

4.  Change  4  pwt.  to  grains.     25  oz.  to  pwt. 

Avoirdupois    Weight. 

364.  Avoirdupois  Weight  is  used  in  weighing  coarse  articles; 

as  liay,  cotton,  groceries,  etc.,  and  all  metals  except  gold  and 

silver. 

Ta  ble. 

16  ounces  (oz.)        =     1     pound,    -    -    -    lb. 

^^^  ,  ,   (  cental,  or     -     -     ctl. 

100  pounds  =     1  i  ,       1     1      •  1  i.  ^ 

■^  ( hundredweight,     ciut. 

2000  lb.,  or  20  cwt.     =     1     ton,    -     -     -    -     T. 

Note.— In  calculating  duties,  etc.,  112  lb.  are  called  a  act.,  and  2240 
lb.  a  long  ton. 

365.  Gross  Weight   is  the  weight   of  goods  including  the 
boxes,  etc.,  which  contain  them. 

Net  Weight  is  the  weight  of  goods  after  deducting  all  allow- 
ances. 


Time. 


155 


366.  Comparison  of  Avoirdupois  and  Troy  Weight. 


7000     grains  Troy 
5760    grains     '' 
43 7|-  grains     " 
480    grains     ^^ 


1  lb.  Avoirdupois. 
1  lb.  Troy. 
1  oz.  Avoirdupois. 
1  oz.  Troy. 


Apothecaries    Weight. 

367.  Apothecaries  Weight  is  used  by  Apothecaries  in  mixing 
medicines.     (Art.  898,  Appendix.) 

Tab  l  e. 

20  grains  {gr.)     =     1  scruple,     -    -     sc,   or  3. 

3  scruples  =     1  dram,  -     -    -    dr.,  or  3. 

8  drams  =     1  ounce,  -     -    -    oz.,    or  3 . 

12  ounces  =     1  pound,      -     -     lb.,    or  lb. 

Note. — The  pound,  ounce,  and  grain  are  the  same  as  Troy  weight. 


Oral    Exerci  s€s. 

1.  How  many  ounces  in  5  pounds  ?    In  100  pounds? 


In  6200  lbs.  ? 
In  ^  ton  ? 


368. 

2.  How  many  tons  in  4000  lbs.  ? 

3.  How  many  pounds  in  ^  ton  ? 

4.  Ac  90  cts.,  what  will  J  lb,  of  tea  cost  ? 

5.  At  $20  a  ton,  what  wiU  J  ton  of  hay  cost  ? 

Time. 

369.  Time  is  a  measured  portion  of  duration, 
are  shown  in  the  following 

Tab  le. 


Its  divisions 


60  seconds  {sec.) 
60  minutes 
24  hours 
7  days 

365  days 

366  days 

12  calendar  months  {mo.) 
100  years 


1  minute,    -  -  min, 

1  hour,  -     -  -  Jir. 

1  day,    -    -  -  d. 

1  week  -     -  -  ivh. 

1  common  year,  c.  yr, 

1  leap  year,  -  I.  yr. 

1  civil  year,  -  yr. 

1  century,  -  -  C. 


156  Compound  Niimnbers. 

370.  A  Civil  Day  is  the  day  adopted  by  government  for 
business  purposes.  It  begins  and  ends  at  midnight,  and  is 
divided  into  two  parts  of  12  hours  each ;  the  former  being 
designated  a.  m.,  the  latter  p.  m. 

371.  The  Solar  Year  is  equal  to  365  d.  5  hr.  48  min.  49.7  sec, 
or  365;}  d.  nearly.  In  4  years  this  fraction  amounts  nearly  to 
1  day.  To  provide  for  this  excess,  1  day "  is  added  to  the  mo. 
of  Feb.  every  4th  year,  which  is  called  Leap  Year.* 

372.  The  Civil  year  includes  both  common  and  leai^  years, 
and  is  divided  into  12  Calendar  months,  viz: 


January 

(Jan.) 

31  days. 

July 

(July) 

31  days 

February 

(Feb.) 

28     " 

August 

(Aug.) 

31     " 

March 

(Mar.) 

31     " 

September 

(Sept.) 

30     " 

April 

(Apr.) 

30     " 

October 

(Oct.) 

31     " 

May 

(May) 

31     " 

November 

(Nov.) 

30     " 

June 

(June) 

30     " 

December 

(Dec.) 

31     " 

Note. — Tlie  following  couplet  will  aid  the  learner  in  remembering  the 
mouths  that  have  30  days  each  : 

"  Thirty  days  hath  September, 
April,  June,  and  November." 

All  the  rest  have  31  days,  except  Fehfuary,  which  in  common  years  has 
28  days  ;  in  leap  years,  29. 

Oral     Exercises. 

373.     1.  How  many  days  in  7  weeks  ?     In  9  weeks  ? 

2.  How  many  weeks  in  42  days  ?     In  63  days  ?   In  90  days  ? 

3.  How  many  months  in  6  years  ?   In  8  years  ?  In  11  years? 

4.  In  48  months  how  many  years  ?     In  72  months  ? 

5.  How  many  centuries  in  500  years  ?     In  1800  years  ? 

6.  At  $9  a  week,  how  much  will  a  man  e[irn  in  6  weeks? 

7.  If  you  pay  ^3  a  week,  how  long  can  you  board  for  $60  ? 

8.  How  many  days  has  a  person  lived  who  is  12  years  old? 

9.  If  you   count   60   a  minute,  how   long  will   it   take  to 
count  1800  ? 

*  For  an  explanation  of  the  mean  Solar  days,  leap  years,  etc.,  see  Art.  901,  Appendix- 


Cf/rcular  Measure,  157 


Circular    Measure. 

374.  Circular  Measure  is  used  in  measuring  angles,  latitude 
and  lo7igitude,  lieavejily  bodies,  etc. 

375.  A  Circle  is  a  plane  figure  bounded  by  a  curve  line 
every  part  of  which  is  equally  distant  from  a  point  within, 
called  the  center. 

376.  The  Circumference  of  a  circle  is 
the  curve  line  by  which  it  is  bounded  ; 
as  ADEBF. 

377.  The  Diameter  is  a  straight  line 
drawn  through  the  center,  terminating  at 
each  end  in  the  circumference  ;  as  AB. 

378.  The  Radius  is  a  straight  line  drawn  from  the  center  to 
the  circumference,  and  is  equal  to  liaJf  the  diameter  ;  as  AC,  DO. 

379.  An  Arc  is  any  part  of  the  circumference ;  as  AD 

Table. 

60  seconds  (")     =     1  minute,     -     -  '. 

60  minutes  =     1  degree,      -     -  °,  or  deg. 

30  degrees  =     1  sign,     -     -     -  S. 

12  signs,  or  360°=     1  circumference,  Cir. 

380.  The  Measure  of  an  angle  is  the  arc  of  a  circle  included 
between  its  two  sides,  as  the  arc  DE. 

The  Standard  Unit  for  measuring  angles  is  the  Degree. 

381.  A  Degree  is  the  angle  measured  by  the  arc  of  -g^  part 
of  the  circumference  of  a  circle. 

The  length  of  the  arc  which  measures  an  angle  of  1°,  varies 
according  to  the  size  of  different  circles,  while  the  angle 
remains  the  same. 

A  degree  at  the  equator,  also  the  average  degree  of  latitude, 
adopted  by  the  U.  S.  Coast  Survey,  is  equal  69.16  miles,  or 
69^  miles,  nearly. 


158  Compouhd  Numbers. 

382.  A  Semi-circumference  {lialf  a  circumference)  is  an  arc 
of  180°,  as  AFB. 

383.  A  Quadrant,  or  one-fourth  of  a  circumference,  is  an  arc 
of  90°,  as  EB. 

A  right  angle  contains  90° ;  for  the  quadrant,  which  meas- 
ures it,  is  an  arc  of  90°. 

Oral     Exercises. 

384.  1.  How  many  degrees  in  J  a  cir.  ?    In  J  cir.  ? 

2.  How  many  degrees  in  a  quadrant?     In  a  right  angle? 

3.  How  many  miles  in  2°  ?    3°  ?    5°  ? 

4.  Through  how  many  deg.  does  the  hour-hand  of  a  clock 
moA  e  in  12  hours  ?     In  3  hrs.  ?     In  6  hrs.  ?     In  1  hr.  ? 

5.  Through  how  many  degrees  does  the  minute-hand  of  a 
clock  pass  in  1  hour  ?     In  \  hr.  ?     In  J  hr.  ?     In  1  minute  ? 

6.  In   making   a  voyage   around   the  world,   through  how 
many  degrees  would  you  sail  ? 


FoREiG-N    Moneys. 

385.  English  or  Sterling  Money  is  the  currency  of  Great 
Britain. 

Table. 

4  farthings  {qr.  ov  far.)  =  1  penny,     -    .    .    .     d. 

12  pence  =  1  shilling,  -     -    -    -    s. 

20  shillings  =  1  pound  or  sovereign,  £. 

10  florins  {fl.)  =  1  pound,    ----£. 

386.  The  Unit  of  English  Money  is  the  Pound  Sterling, 
which  is  represented  by  a  gold  Sovereign  equal  in  value 
to  14.8665. 

387.  Canada  Money  is  expressed  in  dollars,  cents,  and  mills, 
which  have  the  same  nominal  value  as  the  corresponding 
denominations  of  U.  S.  money. 


Foreign  Moneys.  159 

388.  French  Money  is  the  currency  of  France. 

Table. 

10    centimes        =        1     decime. 
10     decimes  =:         1    franc. 

389.  The  Unit  of  French  Money  is  the  Franc,  the  value  of 
which  in  U.  S.  money  is  19.3  cts.,  or  about  i-  of  a  dollar. 

Note. — The  system  is  founded  upon  the  decimal  notation  ;  hence,  all 
operations  in  it  are  the  same  as  those  in  U.  S.  money. 

390.  The  Money  Unit  of  the  German  Empire  is  the  Mark, 
whicli  is  divided  into  100  pennies. 

The  value  of  a  Mark  is  $0,238,  or  %\  nearly.* 

Miscellaneous    Tables. 

12  things  =  1  dozen. 

12  dozen  =  1  gross. 

12  gross  =  1  great  gross. 

20  things  r=  1  score. 

24  sheets    •=  1  quire  of  paper. 

20  quires    =  1  ream. 

2  reams     =  1  bundle. 

5  bundles  =  1  bale. 

2  leaves  =  1  folio. 

4  leaves  =  1  quarto,  or  4to. 

8  leaves  =  1  octavo,  or  8vo. 

12  leaves  =  1  duodecimo,  or  12mo. 

Note. — The  terras  folio,  quarto,  octavo,  etc.,   denote  the  number  of 
leaves  into  which  a  sheet  of  paper  is  folded  in  making  books. 

*  For  Table  of  Foreign  Coins,  see  Art.  631. 


160  Compound  Numbers. 


Oral     Exercises. 

391.     1.   How  many  farthings  in  5  shillings  ? 

2.  How  many  pence  in  £3  ? 

3.  What  cost  8  meters  of  lace,  at  12  francs  a  meter  ? 

4.  How  many  Sovereigns  will  12  yards  of  silk  cost,  at  10s.  d 
yard  ? 

5.  What  will  8  doz.  eggs  cost,  at  a  cent  apiece  ? 

6.  What  will  18  quires  of  paper  cost,  at  20  cts.  a  quire? 

7.  What  will  a  gross  of  buttons  cost,  at  15  cts.  a  dozen  ? 

Questions. 

334.  What  is  a  simple  number?  335.  Compound?  336.  What  is  a 
measure?  337.  For  what  is  linear  measure  used?  338.  What  is  a  line? 
Recite  the  table.     339.  What  is  the  standard  unit  of  length? 

341.  For  what  is  square  measure  used?  342.  What  is  a  surface  ?  343. 
An  angle  ?  The  vertex  ?  344.  A  right  angle  ?  345.  What  is  a  square  ? 
347.  What  is  the  area  of  a  figure  ?    Repeat  the  table. 

349.  For  what  is  cubic  measure  used  ?  350.  What  is  a  solid  ?  351. 
What  is  a  cube?     Recite  the  table. 

356.  For  what  is  liquid  measure  used  ?    Recite  the  table. 

359.  For  what  is  dry  measure  used  ?    Repeat  the  table  ? 

362.  For  what  is  Troy  weight  used?  Recite  the  table.  363.  The 
standard  unit  of  weight  ?  364.  For  what  is  Avoirdupois  weight  used  ? 
Recite  the  table.  What  is  a  long  ton  ?  367.  For  what  is  Apothecaries 
weight  used  ? 

369.  What  is  Time?  Recite  the  table.  370.  What  is  a  civil  day? 
Themeaningof  A.M.  ?  Of  p.m.  ?  371.  Length  of  a  Solar  year  ?  372.  How 
many  calendar  months  in  a  civil  year?     Name  them. 

374.  For  what  is  circular  measure  used?  375.  What  is  a  circle? 
376.  The  circumference?  377.  Diameter?  378.  Radius?  379.  An  arc? 
Table?  380.  The  measure  of  an  angle?  381.  What  is  a  degree?  382. 
A  semi-circumference  ?  383.  A  quadrant  ?  How  many  degrees  in  a  right 
angle  ? 

385.  What  is  English  or  Sterling  money  ?  Repeat  the  table.  386.  The 
iinit  of  English  money?  Its  value?  387.  How  is  Canada  money  ex- 
pressed? 388.  What  is  French  money?  Recite  the  table.  389.  The 
unit  ?  Its  value  ?  390.  What  is  the  money  unit  of  the  German  Empire  ? 
Its  value  ?    Recite  the  miscellaneous  tables. 


EDUCTION. 


Oral     Exercises. 

392.  1.  How  many  pints  in  3  gallons  ? 

Analysis.— In  1  gal.  there  are  4  qt„  and  in  3  gal.,  3  times  4,  or  13  qt. 
In  1  qt.  there  are  2  pints,  and  in  13  qts.,  13  times  3,  or  34  pints,  Ans. 

2.  How  many  feet  in  4  yd.  ?    In  8  yd.  ? 

3.  In  6  sq.  yd.  how  many  square  feet  ? 

4.  How  many  gills  in  9  quarts  ?    In  12  quarts  ? 

5.  In  10  bushels  how  many  pints  ? 

6.  Id  5  days  how  many  minutes  ? 

393.  Keduction  is  changing  Compound  ^Numbers  from  one 
denomination  to  another  without  altering  their  values.  It  is  of 
two  kinds,  Descending  and  Ascending, 

394.  Reduction  Descending  is  changing  higher  denomina- 
tions to  loioer  ;  as,  yards  to  feet,  etc. 

395.  To  reduce  Higher  Denominations  to  Loiver, 

1,  Reduce  34  rods  4  yds.  2  ft.  to  feet. 

Analysis. — As  5|  yds.  make  1  rod,  there  must  34  r.  4  yd.  2  ft. 
be  5|^  times  as  many  yards  as  rods  ;  and  (34  x  5^)  5 1 

+  4  (the   given  yds.)  =  191   yds.      (Art.   208.)  — ^ 

Again,  as  3  ft,  make  1  yd.  there  must  be  3  times  ^^^  J^®" 

as  many  feet   as   yards;    and  (191  x  3)  +  3  (the  3 

given  ft.)  =  575  feet.     Hence,  the  51^5  f^^     Ans 

Rule. — Multiply  the  highest  clenoinination  hy  the 
nuviber  required  of  the  next  lower  to  make  a  unit  of  the 
higher,  ctncl  to  the  product  acid  the  lower  denojiiination. 

Proceed  in  this  manner  with  the  successive  denomina- 
tions, till  the  one  required  is  reached. 


162  Com/pound  Numhers. 

2.  In  5  mi.  12  rd.  4  yd.  2  ft.  liow  many  feet  ? 

3.  Eeduce  143  lb.  3  oz.  6  pwt.  to  grains. 

4.  Reduce  217  tons  35  lb.  to  pounds. 

5.  Reduce  106  tons  68  lb.  to  ounces. 

396.  Reduce  the  following : 

6.  23  mi.  5  rd.  6  ft.  to  feet.  13.  32  A.  6  sq.  rd.  to  sq.  feet 

7.  24  lb.  4  oz.  6  pwt.  to  gr.  14.  26  C.  7  cu.  ft.  to  cu.  ft. 

8.  48  T.  2  cwt.  36  lb.  to  oz.  15.  36  wk.  1  d.  5  hr.  to  min. 

9.  328  gal.  3  qt.  1  pt.  to  gills.  16.  21  yr.  26  d.  to  hours. 

10.  85  hhd.  15  gal.  to  pints.  17.   145°  28"  to  seconds. 

11.  45  bu.  3  pk.  4  qt.  to  pints.       18.   £68  3s.  6d.  to  pence. 

12.  124  sq.  yd.  8  sq.  ft.  to  sq.  in.    19.  £205  7s.  ^(\.  to  far. 

20.  How  many  sec.  in  3  yr.  42  wk.  5  d.  9  hr.  17  min.  ? 

21.  What  will  7  bu.   3  pk.  of  cranberries  cost  at  8  cts.  a 
quart  ? 

22.  Bought  84  gal.  syrup  at  75  cts.  a  gal.,  and  sold  it   at 
22  cts.  a  quart ;  what  was  the  gain  ? 

23.  What   is  the  value  of   12  lb.  5  oz.  6  pwt.  of   gold,  at 
87  cts.  a  pwt.  ? 

Oral     Exercises. 

397.  1.  In   64  pints    how    many    quarts  ?      How    many 
gallons  ? 

Analysis. — Since  in  2  pints  there  is  1  qt.,in  64  pints  there  are  32  quarts. 
In  4  qts.  there  is  1  gallon,  and  in  32  qts.  there  are  8  gallons,  Ans. 

2.  How  many  feet  in  120  inches  ?     How  many  yards  ? 

3.  In  60  ounces  Troy,  how  many  pounds  ?     In  168  ounces  ? 

4.  In  72  hr.  how  many  days  ?     In  96  hours  ? 

5.  How  many  cords  of  wood  in  72  cord  feet  ? 

6.  Change  120s.  to  pounds,  and  240  pence  to  shillings. 

398.  Reduction  Ascending  is  changing  lower  denominations 
to  liiglier  ;  as,  feet  to  yards,  etc. 


Reduction.  163 

399.  To  reduce  Loiver  denominations  to  Higher, 

1.  Reduce  6900  inches  to  rods,  etc. 

Analysis.— Since  12  in.  make  1  ft.,  6900  in.  12  )  6900  in. 

=  as  many  feet  as  12  is  contained  times  in  G900,  "7 

or  575  ft.'   As  3  ft.  make  1  yd.,  575  ft.  =  as  ^  lATr.  -'■'^* 

many  yd.  as  3  is   contained  times  in  575,  or  54  )  191  yd.  2  ft. 

191  yd.  and  2  ft.  over.     Finally,  as  5|yd.  make  o 

1  rod,  191  yd.  =  as  many  rods  as  5i  is  contained  

times  in  191,  or  34  rd.  and  8  half  yards,  or  4  yd.  11  )  382 

over.      (Art.  217.)       Am.    34  rd.  4  yd.  2  ft.  ^  ^^  ^     ^ 

Hence,  the  '     *^ 

Rule. — Divide  the  given  denomination  by  the  ninnher 
reqnii^ed  to  make  one  of  the  Jiext  higher. 

Proceed  in  this  manner  with  the  successive  denomina- 
tions, till  the  one  required  is  reached.  Tlie  last  quotient, 
with  the  several  remainders  annexed,  ivill  he  the  answer. 

Note. — The  remainders  are  the  same  denomination  as  the  respective 
dividends  from  which  they  arise. 

400.  Proof. — Reduction  Ascending  and  Descending  prove 
each  other ;  for,  one  is  the  reverse  of  the  other. 

2.  In  245640  ft.   how  many  miles,  rods,  etc.  ? 

Ans.  46  mi.  4  fur.  7  rd.  1  yd.  1  ft.  6  in. 
Reduce  the  following  to  the  denominations  indicated  : 

3.  34248  gills  to  bbl.  11.   85264  sq.  ft.  to  sq.  rods. 

4.  46840  pt.  to  hhd.  12.   2118165^  sq.  yd.  to  acres. 

5.  653674  pwt.  to  lb.  13.   16568  cu.  ft.  to  cords. 

6.  426508  gr.  to  lb.  14.  43228  qt.  to  bushels. 

7.  35624  oz.  to  cwt.  15.  28956  pt.  to  bushels. 

8.  8420724  oz.  to  tons.  16.  5685720  hr.  to  com.  yr. 

9.  29728  in.  to  rods.  17.  856700  d.  to  weeks. 

10.  48400  ft.  to  miles.  18.  4683248  far.  to  pounds. 

19.  What  will  a  can  of  milk  containing  28  gal.  3  qt.  cost,  at 
6  cts.  a  quart  ? 

20.  At  $0.75  a  yd.,  what  will  it  cost  to  build  a  wall  182  r.  long  ? 

21.  If  a  grocer  buys  3  bu.  of  cranberries  at  $2.25  a  bu.  and 
sells  them  at  9  cts.  a  quart,  how  much  does  he  make  ? 


164  Denominate  Fractions. 


Denominate    Fractions. 

401.  Denominate  Fractions  are  fractions  of  denominate 
Integers,  and  may  be  common  or  decimal. 

402.  To  reduce  Denominate  Fractions,  Common  or  Decimal,  of 
higher  denominations,  to  Integers  of  lower  denominations. 

1.  Eeduce  -J  yard  to  integers  of  lower  denominations. 

Solution.  —  1  yd.  =  8  ft.,   and  -|  yd.  X  3  =  -V-,  or  2|  ft. 

I  yd  X  3  =  -V-  ft-  or  2f  ft.      Again,  |  ft.  X  12  =  ^«S  or  7|  in. 

f  ft.  X 12  =  «/  in.,  or  7i  in.  j^^^^    2  ft.  7^  in. 

2.  Eednce  .875  yard  to  integers  of  lower  denominations. 

Solution.— 1  yd.  =.3  ft.,  and  .875  yd,  x  3  =  2.625  ft.  .875  yd. 

Again,  .625  x  12  =  7.500  inches.  g 

The  answer  is  2  ft.  7.5  inches,  the  same  as  above.  — 

2.625  ft. 

Note. — Pointing  off  3  figures  in  the  several  products  -.  « 

is  equivalent  to  dividing  them  by  1000,  the  denominator  

of  the  given  decimal.     Hence,  the  7.500  in. 

Rule. — Multiply  the  given  iiinnerator,  whether  couzmon 
or  decimal,  and  the  remainder,  if  aihy,  by  the  successive 
numbers  which  will  reduce  a  unit  of  the  given  fraction 
to  the  denomination  required,  and  divide  the  several  pro- 
ducts by  the  given  denominator. 

3.  In  ^  day,  how  many  hours  and  minutes  ? 

4.  In  4:1  week,  liow  many  days,  hours,  etc. 

5.  Eeduce  ff  mile  to  furlongs,  etc. 

6.  Eeduce  f  J  bu.  to  pecks,  quarts,  etc. 

7.  Eeduce  f  sq.  mile  to  acres,  rods,  and  yards. 

8.  Eeduce  -j^-g-  gal.  to  the  fraction  of  a  gill. 

9.  What  part  of  a  pint  is  2-0T  ^^  ^  bushel  ? 

10.  Eeduce  £.4625  to  shillings  and  pence. 

11.  Eeduce  .756  gallons  to  quarts  and  pints. 

12.  Eeduce  .6254  days  to  hours,  minutes,  and  seconds. 

13.  Eeduce  .856  cwt.  to  ounces. 

14.  Eeduce  .7582  of  a  bushel  to  pecks,  etc. 

15.  Eeduce  0.98  rod  to  yards,  feet,  and  inches. 


1.5  pt. 


3.75  qt. 


Reduction.  165 

403.  To  Reduce  Denominate  Integers  or  Fractions  of  lower,  to 
Fractions,  either  Common  or  Decimal,  of  higher  denominations. 

16.  Reduce  7s.  6d.  to  the  common  fraction  of  a  pound. 
Solution.— 7s.  6d.  =  90d.,  and  £1  =  210d.     Now,  £^^^  =  £|,  Ans. 

17.  Reduce  3  quarts  1  pint  2  gills  to  the  decimal  of  a  gallon. 

Solution. — Writing  the  numbers  under  each  '*    ■^*  ^* 

other,  the  lowest  denomination  at  the  top,  we  divide  2 

the  2  gi.  by  4,  and  place  the  quotient  .5  below,  at 
the  right  of  the  next  higher  denomination.     Thus,  ^ 

1.5  pt.  -T-  2  =    .75  qt.,  and  so  on.     Hence,  the  AnS.     .9375   ffal. 

Rule. — Reduce  the  giveiv  compound  jzumher  to  the 
lowest  denoiTbination  mentioned  for  the  numerator,  and 
a  unit  of  the  required  fraction  to  the  same  denomina- 
tion for  the  denominator. 

For  decimcds,  divide  the  given  numhers  as  in  reducing 
integers  to  higher  denominations.     (Art.  399. ) 

Note. — If  the  lowest  denomination  of  the  given  number  contains  a 
fraction,  the  number  must  be  reduced  to  the  ])arts  indicated  by  the 
denominator  of  the  fraction. 

18.  Reduce  f  pint  to  the  fraction  of  a  bu.     (Art.  179,  2°.) 

19.  What  part  of  a  bushel  is  3  pk.  5  qt.  1  pt.  ? 

20.  What  part  of  a  gallon  is  3  qt.  1  pt.  3  gills  ? 

21.  Reduce  9  hr.  15  miu.  12  sec.  to  the  fraction  of  a  week. 

22.  Reduce  15f  gr.  to  the  fraction  of  a  pound  Troj. 

23.  What  part  of  an  acre  is  18|^  square  feet  ? 

24.  Reduce  3  pk.  2  qt.  1  pt.  to  the  decimal  of  a  bushel. 

25.  Change  18  hr.  9  min.  to  the  decimal  of  a  day. 

26.  Change  2  ft.  8  in.  to  the  decimal  of  a  yard. 

27.  Change  8  oz.  7  pwt.  12  gr.  to  the  decimal  of  a  lb.  Troy. 

28.  Change  .4  of  a  pt.  to  the  decimal  of  a  gallon. 

29.  Change  .25  lb.  to  the  decimal  of  a  ton. 

30.  Reduce  2  yr.  3  mo.  18  d.  to  the  decimal  of  a  year. 


166  Ootnpound  Numbers, 

404.  To  find  what  part  one  Compound  Number  is  of  another: 

Reduce  tlie  numhers  to  the  same  denomination,  and  make  the 
numher  denoting  the  imrt  the  numerator,  and  tliat  loitli  luhich 
it  is  compared  the  denominator.     (Arts.  226,  249.) 

31.  What  part  of  2  gal.  3  qt.  1  pt.  is  1  gal.  2  qt.  ? 

32.  What  part  of  4  wk.  2  d.  6  hr.  is  3  d.  12  hours  ? 

33.  What  part  of  15  miles  40  rd.  is  6  mi.  30  rods  ? 

34.  What  decimal  of  4  lb.  2  oz.  12  pwt.  is  6  oz.  8  pwt.  ? 

35.  What  decimal  of  10  bu.  3  pk.  4  qt.  is  4  bu.  1  pk.  5  qt. 

405.  To  Reduce  Metric  to  Common  Weights  and  Measures. 

1.  Reduce  84  decimeters  to  feet. 

OPERATION. 

ANALYSis.—Taking  39.37  in.,  the  value  of  the  39.37  m. 

principal  metric  unit,  as  the  standard,  we  multiply  §4  -^^ 

it    by  the  given  metric  number  expressed  in  the  

same  metric  unit ;  and  84  dm.  =  8.4  m.  ^*^  ' '*" 

Since   1  m.  is  equal  to   39.37  in.,    8.4  m.  are  31496 

equal  to  8.4  times  39.37  in.,  or  330.708  in.,  and  -.^^  qon  wao  -^ 

330.708  in.  =  27.559  ft.,  Aiis.     Hence,  the  j  66K).jKm  m. 

Ans.  27.559  ft. 

Rule.  — Multiply  the  value  of  the  prineipal  iivetrie  unit 
of  the  Table  by  the  given  metric  number  expressed  in  the 
same  unit,  and  reduce  the  product  to  the  denomination 
required.     (Art.  399.) 

2.  In  45  kilos,  how  many  pounds?  Ans.  99.207  lb. 

3.  In  63  kilometers,  how  many  miles  ? 

4.  Reduce  75  liters  to  gallons. 

5.  Reduce  56  dekaliters  to  bushels. 

6.  Reduce  120  grams  to  ounces. 

7.  Reduce  137.75  kilos  to  pounds. 

8.  In  36  ars,  how  many  square  rods? 

Analysis. — In  1  ar  there  are  119.6  sq.  yd.  ;  hence  in  36  ars  there  are 
36  times  as  many.  Now  119.6  x  36  =  4305.6  sq.  yd.,  and  4305.6  sq.  yd.-r- 
30^  =  142.33  sq.  rods,  Ans. 

9.  In  60.25  hektars,  how  many  acres  ? 
10.  In  120  cu.  meters,  how  many  cu.  feet  ? 


Addition.  167 

406.  To  reduce  Common  to  Metric  Weights  and  Measures, 

11.  Reduce  2190  yds.  2  ft.  11  in.  to  kilometers. 

OPERATION. 

Explanation. — Reducing  the  omn     i    no.    -• -i  • 

,      ^    .    ,  ,  2190  yd.  2  ft.  11  in. 

given  number  to  inches  we  have  "^ 

7887o  in.     Dividing  this  number  £ 

by  89.37,  the  number  of  inches  in  6572  ft. 

a  meter,  we  have  2003.429+  m.  ..« 

To    reduce    meters    to   Km.    we  

remove  the  decimal  point  3  places  39.37  )  78875  in. 

to  the  left.    Ans.  2.003429  +  Km. 

Hence,  the 


2003.429+  m. 
A71S.  2.003429+  Km. 


Rule. — Divide  the  given  number  by  the  value  of  the 
principal  jnetrie  unit  of  the  Table,  and  reduce  the  quo- 
tient to^the  denomination  required. 

Note. — Before  dividing,  the  given  number  should  be  reduced  to  the 
denomination  in  which  the  Dalue  of  the  principal  unit  is  expressed. 

12.  In  63f  yards,  how  many  meters  ? 

13.  Reduce  13750  pounds  to  kilograms. 

14.  Reduce  250  liquid  quarts  to  liters. 

15.  Reduce  2056  bu.  3  pecks  to  kiloliters. 

16.  In  3  cwt.  15  lb.  12  oz.,  how  many  kilos? 

17.  In  7176  sq.  yards,  how  many  sq.  meters  ? 

18.  In  40.471  acres,  how  many  hektars  ? 

19.  In  14506  cu.  feet,  how  many  cu.  meters  ? 

20.  In  36570  cu.  yards,  how  many  cu.  meters? 


Addition. 

407.  The  method  of  Adding,  Subtracting,  Multiplying,  and 
Dividing  Compound  Numbers  is  the  same  as  the  correspond- 
ing operations  in  simple  numbers  and  special  rules  are  unnec- 
essary. 

Note  — 1.  The  apparent  difference  arises  from  their  scales  of  increase, 
one  being  variable  and  irre^lar,  the  other  decimal  and  uniform. 


168 


Compound  Nuinbers, 


1.   What  is  the  sum  of  18  bu.  3  pk.  5  qt.  1  pt.,  24  bu.  2  pk. 
6  qt.,  6  bu.  2  pk.  7  qt.  1  pt,  8  bu.  3  pk.  4  qt.  1  pt.  ? 


Explanation.  —  The  sum  of  the  right- 
hand  col.  is  3  pt.  =  1  qt.  1  ])t.  Set  the  1  pt. 
under  the  col.  of  pints,  and  adding  the  1  qt. 
to  the  col.  of  qt.,  the  sum  is  23  qt.  =  2  pk. 
7  qt.  Write  the  7  qt.  in  the  col.  of  quarts,  and 
adding  the  2  pk.  to  the  col.  of  pk.,  proceed  as 
before.     Ans.  59  bu.  0  pk.  7  qt.  1  pt. 


bu. 
18 
24 

6 

8 


OPEKATION. 

pk.      qt. 

3       5 


59       0 


pt. 
1 

1 
1 

1 


(2.) 


£ 
5 
6 
5 
12 


far. 
3 

2 
1 
2 


(3.) 
gal.     qt.     pt. 

2  1 

3  0 
0  1 
3       1 


2 
1 

2 
0 


(4.) 

wk.    da.      hr.  min. 

2       3       8  40 

4       6       5  10 

2       5      20  35 

6       4     18  23 


5.  What  is  the  sum  of  5  rd.  4  yd.  2  ft.  7  in.,  6  rd.  5  yd.  2  ft. 
6  in.,  4  rd.  4  yd.  0  ft.  4  in.,  3  rd.  3  yd.  2  ft.  8  in.  ? 


Note. — 2.  When  a  fraction  occurs  in  the 
amount  in  any  denomination  except  the  lowest, 
it  should  be  reduced  to  integers  of  lower  de- 
nominations, and  uuited  with  like  integers. 
Thus,  in  Ex.  5  the  \  yd.  =  1  ft.  6  in.,  which 
added  to  2  ft.  1  in.  make  3  ft.  7  in. ;  and  1  yd. 
plus  3  ft.  plus  7  in.  equals  2  yd.  0  ft.  7  in. 


rd. 

yd- 

ft. 

in. 

5 

4 

2 

7 

6 

5 

2 

6 

4 

4 

0 

4 

3 

3 

2 

8 

21 

H 

2 

1 

i  = 

:   1 

G 

Ans.  21 


0       7 


6.  What  is  the  capacity  of  3  bins  holding  respectively  35  bu. 
3  pk.  4  qt.,  42  bu.  1  pk.  6  qt.,  and  56  bu.  2  pk.  5  qt.  ? 

7.  How  mucli  land  in  3  farms  containing  87  A.  48  sq.  rd., 
97  A.  67  sq.  rd.,  and  65  A.  42  sq.  rd.  ? 

8.  Bought  3  casks  of  oil ;  holding  2  hhd.  30  gal.  2  qt. ;  3  hhd. 
10  gal.;  1  hhd.  13  gal.  1  qt.;  how  much  did  all  hold  ? 

9.  Add  together  23  yr.  2  mo.  3  wk.  5  d.,  68  yr.  3  mo.  2  wk. 
3  da.,  60  yr.  4  mo.  1  wk.  6  d.,  49  }t.  and  4  d. 

10.  Required  the  number  of  miles,  etc.  in  3  roads,  measuring 
23  mi.  67  rd. ;  32  mi.  65  rd.  ;  and  46  mi.  28  rods. 


Subtraction,  169 

11.  A  mason  plastered  one  room  containing  45  sqtiare  yards 
7  ft.  6  in.,  another  25  sq.  yd.  6  ft.  95  in.,  another  38  sq.  yd.  4  ft. 
41  in.;  Avhat  was  the  amount  of  his  phistering? 

12.  One  pile  of  wood  contains  10  C.  38  ft.  39  in.,  another 
15  C.  56  ft.  73  in.,  another  30  C.  19  ft.  44  in.,  another  17  0. 
84  ft.  21  in.;  how  much  do  they  all  contain? 

13.  Find  the  sum  of  45  mi.  17  rd.  5  yd.  %  tt.  9  in.,  43  mi. 
44  yd.  1  ft.  8  in.,  89  mi.  216  rd.  3  yd.  2  ft.  5  in. 

14.  What  is  the  sum  of  £  J,  \^. ,  and  ^d.  ? 


OPERATION. 


Note.— 3.  Denominate  Fractions  should  ^   ~  ^*       '  ^    ^^^' 

be  reduced  to  integers  of  lower  denomina-  i^^*   -—  ^^*  IQ-*  -^     l^-r. 

tions,  then  added  as  above.     (Art.  402.)  Jd.  =:  Os.  Od.  1^  far. 


Ans. 

3s.  ^ 

id.  3^  far. 

15. 

Add 

1  hu. 

H 

pk. 

|qt.  i 

pt.) 

■A 

bu.  \  pk. 

f  qt. 

■ipt. 

16. 

Add 

1  of 

A 

day,  f  of 

A  h 

I--, 

A  of  M 

min. 

,  and  f  of 

2^  sec. 

17. 

Add 

|lb.  1 

to| 

:  oz. 

1  pwt. 

19. 

f  gal.  to 

iqt. 

lipt. 

18. 

Add 

fwk. 

to 

1  d. 

1|  hr. 

20. 

£-1  to  f  s. 

2|d. 

Subtraction. 

408.     1.  From  35  rd.  2  yd.   1  ft.   8  in.,  take  22  rd.  2  yd. 
2  ft.  6  in. 

Explanation.  —  Write  the 
numbers  and  proceed  as  in  sim- 
ple subtraction.  Taking  6  in. 
from  8  in.  leaves  2  inches.  As 
2  ft.  cannot  be  taken  from  1  ft.,  AUS. 
SVC  take  1  yd.  =  8  ft.  from  2  yd., 
and  adding  it  to  1  ft.,  we  have  Qj. 
4  ft.,  and  2  ft.  from  4  ft.  leave  2  ft. 

Again,  2  yd.  cannot  be  taken  from  the  1  yd.  remaining, 
yd.,  added  to  1  yd.  make  6|  yd.,  from  which  subtract  2  yd.,  and  A.\  yd. 
remain.  Finally,  22  rd.  from  34  rd.  leave  12  rd.  The  4  yd.  =  1  ft.  6  in., 
which  added  to  the  above  make  12  rd.  5  yd.  0  ft.  8  in.,  Ans. 

2.  From  121  hhd.  28  gal.  1  qt.,  take  63  hhd.  21  gal.  3  qt. 
8 


OPERATION. 

35  rd.     2  yd.     1  ft. 
22           2           2 

8  in. 
6 

12           ^         2 
i   -   1 

2 
6 

12           5            0 

'd.  remaining.      But  1  re 

8 
1.  =5^ 

170  Compoimd  Namhers. 

3.  Bought  2  silver  pitchers,  one  weighing  2  lb.  10  oz.  10  pwt. 
1  gr.,  the  other  2  lb.  3  oz.  12  pwt.  5  gr. ;  what  is  the  difference 
in  their  weight  ? 

4.  A  merchant  had  228|  yards  of  cloth,  and  sold  115|  yards; 
how  much  had  he  left  ? 

5.  From  25  mi.  7  fur.  8  rd.  12  ft.  6  in.,  take  16  mi.  6  fur.  30 
rd.  4  ft.  8  in. 

6.  A  man  owning  95  A.  75  rd.  67  sq.  ft  of  land,  sold  40  A. 
86  rd.  29  ft. ;  how  much  had  lie  left  ? 

7.  A  tanner  built  two  cubical  vats,  one  containing  116  ft. 
149  in.,  the  other  245  ft.  73  in.;  what  is  the  difference  be- 
tween them  ? 

8.  A  man  having  65  0.  95  ft.  123  in.  of  wood  in  his  shed, 
sold  16  C.  117  ft.  65  in. ;  how  much  had  he  left? 

409.   To    find    the    Exact    Number    of  Years,  Months,  and  Days, 

between  two  dates. 

9.  What  is  the  difference  of  time  between  July  4th,  1879, 
and  Nov.  15th,  1882  ? 

Analysis.— The  time  from  July  4th,  1879  to  July  4th,  1882  =  3  yr. 
The  time  from  July  4tli  to  Nov.  4th  —  4  mo 

The  time  from  Nov.  4th  to  Nov.  15th.  =  11  d. 

Ans.  3  yr.  4  mo.  11  d.     Hence,  the 

Rule. — First  find  the  nmiiher  of  entire  years,  next  the 
ninnher  of  entire  months,  then  the  days  in  the  parts  of  a 
month. 

Note. — 1.  The  day  on  which  a  note  or  draft  is  dated,  and  that  on 
which  it  becomes  due,  must  not  both  be  reckoned.  It  is  customary  to  omit 
the  former  and  count  the  latter. 

10.  A  ship  started  on  a  trading  voyage  round  the  world  Mar. 
3d,  1875,  and  arrived  back  Aug.  24th,  1878 ;  how  long  was 
she  gone  ? 

11.  What  is  the  time  from  Oct.  15th,  1875,  to  March  10th, 
1882? 

la.  A  note  dated  Oct.  2d,  1870,  was  paid  Dec.  25th,  1882 ; 
how  long  was  it  from  its  date  to  its  payment  ? 

13.  A  ship  sailed  on  a  whaling  voyage,  Aug.  25th,  1880,  and 
returned  April  15th,  1882  ;  how  long  was  she  gone  ? 


Oct. 

=  31  d, 

Nov. 

=  30d, 

Dec. 

=  31d. 

Jan. 

=  15  d. 

Ans.  119  d. 

Subtraction.  171 

14.  A  mortgage  was  dated  April  10th,  1875,  and  was  paid 
Aug.  25,  1880  ;  how  long  did  it  run  ? 

15.  How  many  days  did  a  note  run  which  was  dated  Sept. 
18th,  1879,  and  paid  Jan.  15th,  1880  ? 

Analysis.— In  Sept.  it  had  30-18=12  operation. 

days;  in  Oct.,  31  d.  ;  in  Nov..  30  d. ;  in  Sept.  30  —  18  =  12  d. 

Dec,  31  d. ;  in  Jan.,  15  d.     Hence, 

Note. — 2.  To  find  the  number  of  days 
between  two  dates,  write  in  a  col.  the  num- 
ber of  days  remaining  in  the  first  mo., 
and  the  number  in  each  succeeding  month, 
including  those  in  the  last ;  the  sum  will 
be  the  number  of  days  required. 

16.  A  note  dated  May  21st,  1879,  was  paid  Nov.  28th,  1879 ; 
how  many  days  did  it  run? 

17.  What  is  the  number  of  days  between  Oct.  5th,  1879,  and 
March  3d,  1880  ? 

18.  A  person  started  on  a  journey  Aug.  19th,  1869,  and 
returned  Nov.  1st,  1869  ;  how  long  was  he  absent  ? 

19.  A  note  dated  Jan.  31st,  1870,  was  paid  June  30th,  1870  ; 
how  many  days  did  it  run  ? 

20.  How  many  days  from  May  23d,  1868,  to  Dec.  31st,  fol- 
lowing ? 

21.  The  latitude  of  New  York  is  40°  42'  43''  N.,  that  of  St. 
Augustine,  Fla.,  is  29°  48'  30"  ;  what  is  the  difference  of  their 
latitude  ? 

Note.— 3.  When  two  places  are  on  opposite  sides  of  the  Equator,  the 
difference  of  latitude  is  found  by  adding  their  latitudes, 

22.  The  latitude  of  Cape  Horn  is  55°  59'  S.,  that  of  Cape 
Cod  is  42°  1'  57"  N. ;  what  is  the  difference  of  their 
latitude  ? 

23.  The  longitude  of  Cambridge,  Mass.,  is  71°  7'  22",  that 
of  St.  Louis  is  90°  15'  16"  ;  what  is  the  difference  of  their 
longitude  ? 

24.  The  Ion.  of  Paris  is  2°  20'  E.,  that  of  Washington  D.  C. 
is  77°  0'  15"  W. ;  what  is  the  difference  of  their  longitude  ? 


172  Compound  Numhers, 


Multiplication. 

410.  1.  If  a  man  can  build  a  fence  12  rd.  1  yd.  2  ft.  5  in. 
long  in  one  day,  how  long  a  fence  can  he  build  in  6  days  ? 

Analysis.— In  6  d.  he  can  build  12  rd.      1  yd.      2  ft.      5  in. 

6  times  as  much  as  in  1  d.     6  times  g 

5  in.  are  30  in.  =2  ft,  6  in.     Write  -; -^ — -^r~ 

the  G  under  the  in.,  and  add  the  '^^  ^<^-      "^2"  1^'    ^  ^^'       ^  ^^• 

2  ft.  to  the  next  product.     Proceed  (-g-)    =    1            6 

in  this  way  till  all  the  denomina-    ^^ic.    73  y^       5  \i^       \  ft       0  in 
tions  are  multiplied. 

Note. — If  a  fraction  occurs  in  the  product  of  any  denomination  except 
the  lowest,  it  should  be  reduced  to  lower  denominations ,  and  be  united  to 
those  of  the  same  name  as  in  Compound  Addition. 

2.  Multiply  8  lb.  6  oz.  3  pwt.  by  8. 

3.  Multiply  27  gal.  3  qt.  1  pt.  3  gi.  by  7. 

4.  Multiply  26  mi.  87  rd.  4  yd.  2  ft.  by  9. 

5.  What  is  the  weight  of  12  silver  cups,  each  weighing  8  oz. 
17  pwt.  6  gr.  ? 

6.  How  much  water  in  28  casks,  each  containing  54  gal. 

3  qt.  1  pt.  2  gi.  ? 

7.  If  a  railroad  car  goes  21  mi.  2  fur.  10  rd.  per  hour,  how 
far  will  it  go  in  25  hours  ? 


Division. 

411.     1.  A  grocer  paid  £5  2s.  9d.  for  4  boxes  of  sugar ;  how 
much  was  that  a  box  ? 

Analysis.— Since  4  boxes  cost  £5  2s.  9d. ,  operation. 

1  box  will  cost  \  as  much,  and  £5-^4  =  £1  and  4  )  £5      2s.      9d. 

£1  over.     Reducing  the  remainder  to  the  next        j  o-i     ^k7       ^X(\ 

lower  denomination,  and    adding  the  2s.,  we  *  •         4    • 

have  22s.,  which  divided  by  4  =  5s.  and  2s.  over.     Reducing  2s.  as  before, 
continue  the  division  till  each  denomination  is  divided. 

2.  A  silversmith  melted  up  2  lb.  8  oz.  10  pwt.  of  silver, 
which  he  made  into  6  spoons ;  what  was  the  weight  of  each  ? 


Longitude  and  Time.  173 

3.  If  8  persons  consume  85  lb.  12  oz.  of  meat  in  a  month, 
how  much  is  that  apiece  ? 

4.  A  man  traveled  50  mi.  and  32  rd.  in  11  hours;  at  what 
rate  did  he  travel  per  hour  ? 

5.  A  man  had  285  bu.  3  pk.  6  qt.  of  grain,  which  he 
wished  to  carry  to  market  in  15  equal  loads  ;  how  much  must 
he  carry  at  a  load  ? 


Longitude    and    Time. 

412.  The  Earth  turns  on  its  axis  once  in  24  hours  ;  hence, 
^  part  of  360°,  or  15°  of  longitude,  passes  under  the  sun  in 
1  hour. 

Again,  -^  of  IS'^  Ion.,  or  15',  passes  under  the  sun  in  1  min. 
of  time.  And  -^  of  15',  or  15''  Ion.,  passes  under  the  sun  in 
1  sec.  of  time,  as  seen  in  the  following 

413.  Comparison  of  Longitude  and  Time. 

360°  Ion.  make  a  difference  of  24  hrs.  of  time. 

15°     "        ''  "  1  hr.         '' 

1°     "         "  ''  4  min.      '' 

V     ''        ''  ''  4  sec.       " 

1"    ''        "  "         -^sec.       '' 

414.  The  Longitude  of  a  place  is  the  number  of  deg.,  min., 
and  sec,  reckoned  on  the  equator,  between  a  standard  meridian 
(marked  0°)  and  the  meridian  of  the  given  place. 

All  places  are  in  East  or  West  longitude,  according  as  they 
are  East  or  West  of  the  Standard  Meridian,  until  180°,  or  half 
the  circumference  of  the  Earth  is  reached. 

Notes. — 1.  The  Enprlish  reckon  Ion.  from  the  meridian  of  Greenwich  ; 
the  French  from  that  of  Paris.  Americans  generally  reckon  it  from  the 
meridian  of  Greenwich  ;  sometimes  from  that  of  Washington. 

2.  When  two  places  are  on  opposite  sides  of  the  Standard  Meridian, 
the  difference  of  Ion,  is  found  by  adding  their  longitudes.  (Art.  409,  N  3.) 


174  Compound  Numhers. 

415.  To  find   the    Difference    of  Longitude    between  two  places, 

the  Difference  of  Time  being  known. 

1.  The  difference  of  time  between  New  York  and  Chicago  is 
54  min.  19  sec.     What  is  the  difference  of  longitude  ? 

Analysis.  —  Since  15'    of  Ion.  make  a  operation. 

difference  of  1  min.  of  time,  tliere  must  be  54  m.    19  sec. 

15  times  as  many  min.  of  Ion.  as  tliere  are  15 

min.  and  sec.  of  time,  and  (54  min.  19  sec.)  -—      -"-  —,, 

xl5  =  13°84'45",  ^/?«.     Hence,  the  ^^       ^^  45  ,  ^/^S. 

Rule. — Multiply  the  difference  of  time,  expressed  in 
hours,  minutes,  and  seconds,  hy  15 ;  the  product  will  he 
the  diff'erence  of  longitude  in  degrees,  minutes,  and 
seconds.     (Art.  412.) 

2.  The  difference  of  tiaie  between  Boston  and  Albany  is 
9  min.  2  sec. ;  what  is  the  difference  of  longitude  ? 

3.  The  difference  of  time  between  Savannah,  Ga.,  and  Port- 
land, Me.,  is  43  min.  32.13  sec. ;  what  is  the  dif.  of  longitude  ? 

4.  The  difference  of  time  between  Boston  and  Detroit  is 
47  min.  56  sec. ;  w^hat  is  the  difference  of  longitude  ? 

5.  The  difference  of  time  between  Philadelphia  and  Cincin- 
nati is  37  min.  8.4  sec;  what  is  the  difference  of  longitude ? 

6.  The  difference  of  time  between  Louisville,  Ky.,  and  Bur- 
lington, Vt.,  is  49  min.  20  sec;  what  is  the  dif.  of  longitude  ? 

416.  To   find   the   DiflTerence   of  Time   between   two   places,  the 

Difference  of  Longitude  being  known. 

1.  The  difference  of  longitude  between  Chicago  and  Boston 
is  IQ'^  34'  15"  ;  what  is  the  difference  of  time  ? 

Analysis. — Since  15°  Ion.  make  a  operation. 

difference  of  1  hour  of  time,  there  must         15  )  16°        34'  15" 

be  A  as  many  hours    minutes,  and         J,,,,    l  h^.     6  min.  17  sec 

seconds,  as  there  are  deg.,  mm.,   and 

sec.  of  Ion.,  and  (16°  34'  15")  -4- 15  ==  1  hr.  6  min.  17  sec.     Hence,  the 

Rule. — Divide  the  difference  of  longitude,  in  degrees, 
minutes,  and  seconds,  hy  15 ;  the  quotient  will  he  the 
difference  of  time  in  hours,  minutes,  and  seconds. 


Longitude  and  lime.  175 

2.  The  difference  of  longitude  between  Cambridge,  Mass., 
and  Charlottesville,  Va.,  is  T  23'  49" ;  what  is  the  difference 
of  time  ? 

3.  The  Ion.  of  St.  Louis  is  90°  15'  15",  that  of  Charleston, 
S.  C,  is  79°  55'  38"  ;  what  is  the  difference  of  time  ? 

4.  The  Ion.  of  Berlin  is  13°  23'  45"  E.,  that  of  :N^ew  Haven, 
Ct,  is  72°  55'  24"  W. ;  what  is  the  difference  of  time  ? 

5.  The  Ion.  of  Montreal  is  73°  25'  W.,  that  of  New  Orleans  is 
90°  2'  30"  W.  ;  what  is  the  difference  of  time? 

6.  The  Ion.  of  Paris  is  2°  20'  E.,  Kome  is  12°  27'  E. ;  what  is 
the  difference  of  time  ? 

7.  The  Ion.  of  West  Point  is  73^  57'  W.,  that  of  Washington, 
D.  C,  77^  0'  15"  W. ;  what  is  the  difference  of  time? 

8.  How  much  earlier  does  the  sun  rise  in  Albany,  Ion.  73° 
44' 50",  than  in  St.  Paul,  Min.,  Ion.  93°  4'  55"?  Than  in 
Astoria,  Oregon,  Ion.  124°  ? 

9.  When  it  is  9  a.m.  in  New  York,  Ion.  74°  3',  what  is  the 
time  in  Richmond,  Va.,  Ion.  77°  25'  45"  ?  In  San  Francisco, 
Ion.  122°  26'  45"  ? 

Q  U  ESTI ON  S. 

393.  What  is  reduction  ?  393.  Descending?  896.  Rule?  397.  Reduc- 
tion Ascending:  ?    399.  Rule  ?     How  proved  ? 

401.  What  is  a  denominate  fraction  ?  402.  How  reduce  them  from 
higher  denominations  to  integers  of  lower  ?  403.  How  reduce  denominate 
integers  to  fractions  of  higher  denominations  ? 

404.  How  find  what  part  one  number  is  of  another  ?  405.  How  reduce 
metric  to  common  weights  and  measures  ?  406.  How  reduce  common  to 
metric  weights  and  measures  ? 

407.  How  are  compound  numbers  added,  subtracted,  multiplied,  and 
divided  ?  From  what  does  the  apparent  difference  atise  ?  409.  How  find 
the  difference  between  two  dates  in  years,  months,  and  days  ?  How  find 
the  difference  of  latitude  between  two  places  on  opposite  sides  of  the 
equator  ? 

414.  What  is  the  longitude  of  a  place  ?  When  is  a  place  in  East 
longitude  ?  When  in  West  ?  From  what  meridian  do  the  English  reckon 
longitude  ?     The  French  ?     Americans  ? 

415.  How  find  the  difference  of  longitude  when  the  difference  of  time 
is  given  ?  416.  How  find  the  difference  of  time  when  the  difference  of 
longitude  is  given  ? 


176 


Coinpound  Number So 


Measurement    of    Surfaces. 

Oral     Exercises. 

417.     1.  How  many  sq.  feet  in  the  surface  of  a  blackboard 
4  ft.  long  and  3  ft.  wide  ? 


Analysis. — Let  the  sides  of  the  black- 
board be  divided  into  4  equal  parts,  and 
the  ends  into  3  equal  parts,  each  denoting 
a  linear  foot.  The  blackboard  contains 
as  many  sq.  feet  as  there  are  squares  in 
the  figure.  Since  there  are  4  squares  in 
1  row,  in  3  rows  there  are  3  times  4,  or  12 
squares.     Ans.  12  sq.  feet. 


2.  How  many  sq.  feet  in  a  flagging  stone  8  ft.  long  and  4  feet 
wide  ? 

3.  How  many  sq.  feet  in  a  strawberry  bed  20  ft.  long  and 
5  ft.  wide? 

4.  How  many  sq.  yards  in  a  lawn  whose  length  is  9  yards  and 
its  breadth  7  yards  ? 

5.  If  a  meadow  is  12  rods  long  and  8  rods  wide,  how  many 
sq.  rods  does  it  contain  ? 

6.  A  house  lot  containing  84  sq.  rods  is  7  rods  wide  ;  what  is 
its  length  ? 

7.  I  wish  to  lay  out  an  orchard  12  rods  in  width  ;  what  must 
be  its  length  to  contain  240  sq.  rods  ? 

8.  What  is  the  difference  between  4  square  feet  and  4  feet 
square  ? 

Written     Exercises. 

418.  A  Plane  Figure  is  one  which  repre- 
sents a  plane  or  flat  surface. 

419.  The  Perimeter  of  a  plane  figure  is 
the  line  which  hounds  it. 


420.  The  Area  of  a  plane  figure  is  the  quantity  of  surface  it 
contains. 


Measurement  of  Surfaces.  177 

421.  A  Rectangle  is  a  plane  figure  haying  four  sides  and 
four  right-angles.     (Art.  418.) 

422.  When  all  the  sides  of  a  rectangle  are  equal  it  is  called 
a  Square. 

423.  The  Dimensions  of  a  rectangular  figure  are  its  length 
and  Ireadth. 

424.  To  find  the  Area  of  Rectangular  Surfaces. 

1.  How  many  square  rods  in  a  garden  18  rods  long  and 
12  rods  wide  ? 

^  .  ,  -I      -.o  1      1  1     H  J  OPERATION. 

Solution. — A  rectangle  18  rods  long  and  1  rod 
wide  will  contain  18  sq.  rods.    And  a  garden  18  rods  roas. 

long  and  13  rods  wide  will  contain  12  times  18,  or  12 

216  sq.  rods,  Ans.     Hence,  tlie  j^^^^  ^  ^^   ^^^^^^ 

EuLE.- — Multiply  the  length  dy  the  hreaclth. 

Notes. — 1.  Both  dimensions  sliould  be  reduced  to  the  same  denomina- 
tion before  they  are  multiplied. 

2.  One  line  is  said  to  be  multiplied  by  another,  when  the  rivmber  of 
units  in  the  former  are  taken  as  many  times  as  there  are  like  units  in  the 
latter.     (Art.  83,  i°.) 

3.  The  area  and  one  side  of  a  rectangular  surface  being  given,  the  other 
side  is  found  by  dividing  the  area  by  the  gimn  side.     (Art.  119a.) 

2.  How  many  yards  of  carpeting  1  yd.  wide  will  it  take  to 
cover  a  floor  22  ft.  long  and  15  ft.  wide  ? 

3.  How  many  yards  of  carpeting  27  in.  wide  will  it  take  to 
cover  the  same^oor. 

4.  In  a  meadow  68  rd.  long  and  43  rd.  wide,  how  many  acres  ? 

5.  A  building  lot  is  50  ft.  front,  and  contains  half  an  acre  ; 
how  far  back  does  it  extend  ? 

6.  At  25  cts.  per  sq.  foot,  what  is  the  cost  of  an  acre  of  land  ? 

7.  Bought  a  rectangular  farm  210  rods  long  and  88  rods 
wide,  at  $15  per  acre  ;  what  was  the  cost  ? 

8.  The  length  of  a  pasture  is  231  meters,  and  its  breadth  is 
87  meters  :  what  is  its  area  in  sq.  meters  ? 

9;  The  area  of  a  meadow  is  210.6  sq.  meters,  and  its  length 
is  64.8  meters ;  what  ^s  its  width  ? 


178  Compound  Nunribers, 

10.  If  I  pay  1276  for  92  meters  of  broadcloth  1.5  meters  wide, 
what  is  that  per  square  meter  ? 

11.  How  many  acres  in  a  field  800  rods  long,  and  128  rods 
wide  ? 

12.  Find  the  area  of  a  square  field  whose  sides  are  65  rods  in 
length. 

13.  A  man  fenced  off  a  rectangular  field  containing  3750  sq. 
rods,  the  length  of  which  was  75  rods  ;  what  was  its  breadth? 

14.  How  many  hektars  in  a  rectangular  field  475.5  meters 
long  and  246  meters  wide  ? 

15.  The  length  of  the  Capitol  at  Washington  is  751  ft.,  its 
width  348  ft. ;  how  many  sq.  rods,  and  how  many  acres  does  it 
cover  ? 

16.  What  is  the  difference  between  two  asparagus  beds  one 
of  which  is  2  rods  square,  and  the  other  contains  2  sq.  rods  ? 

17.  The  length  of  the  main  Centennial  building  in  Philadel- 
phia was  1880  ft.,  and  the  width  464  ft.  ;  how  many  acres  did 
it  cover  ? 

18.  A  speculator  bought  50  acres  of  land  at  150  per  acre,  and 
sold  it  in  villa  lots  of  5  rods  by  4  rods,  at  1150  a  lot;  what  did 
he  make  by  the  operation  ? 

19.  A  garden  27  yd.  long  and  15  yd.  wide  has  a  gravel  walk 
round  it  6  feet  wide ;  what  did  the  walk  cost,  at  50  cts.  per 
square  yard  ? 

20.  What  will  it  cost  to  carpet  a  floor  18  by  16  ft.,  the  carpet 
being  27  in.  wide,  and  its  cost  $1.12  a  yard  ? 

21.  What  is  the  cost  of  paving  a  street  628  ft.  long  and  60|- 
ft.  wide,  at  12.25  a  sq.  yard  ? 

22.  How  many  tiles  10  in.  square  are  required  to  lay  a  side 
walk  168  ft.  long  and  h\  ft.  wide  ? 

23.  What  will  it  cost  to  concrete  a  court  168  ft.  square,  at 
$3.75  per  sq.  yard  ? 

24.  A  farm  containing  150  acres,  is  200  rods  long ;  what  is 
its  width  ?  What  will  it  cost  to  build  a  wall  around  it,  at  $4  a 
rod? 

25.  How  many  planks  15  ft.  long  and  6  in.  wide  will  it  take 
to  floor  a  room  20  ft.  long  and  15 J  ft.  wide  ? 


Measurement  of  Solids, 


179 


Measurement    of    Solids. 


425.  A  Rectangular  Body  is  one  bounded  by  six  rectangular 
sides,  etich  opposite  pair  being  equal  and  'parallel ;  as,  boxes  of 
goods,  blocks  of  hewn  stone,  etc. 

426.  When  all  the  sides  are  equal,  it  is  a  Cube ;  when  the 
oiJliosite  sides  only  are  equal,  it  is  a  Parallelepiped. 

427.  The  Contents  or  Volume  of  a  body  is  the  quantity  of 
matter  or  siiace  it  contains. 

428.  The  Dimensions  of  a  rectangular  body  are  its  length, 
breadth,  and  thickness. 

429.  To  find  the  contents  or  volume  of  Rectangular  Bodies. 

1.  How  many  cubic  feet  in  a  block  of  granite  4  ft.  long,  3  ft. 
wide,  and  2  ft.  thick  ? 

Illustration. — Let  the  block  be  repre- 
sented by  the  adjoining  figure,  the  length  of 
which  is  divided  into  4  equal  parts,  the 
width  into  3,  and  the  thickness  into  2  pnrts, 
each  of  which  is  a  linear  foot.  Since  the 
block  is  4  ft.  long  and  3  ft.  wide,  in  the 
upper  face  there  are  3  times  4,  or  13  sq.  feet.  Now,  if  the  block  were 
1  foot  thick  it  must  have  as  many  cu.  feet  as  there  are  sq.  feet  in  the 
upper  face.  But  the  given  block  is  2  ft.  thick;  therefore,  it  contains  2  times 
(4  X  3),  or  24  cu,  feet,  Ans.     Hence,  the 

Rule. — Multiply  the  length,  hreaclth,  and  thiclcness 
together.     (Art.  424.) 

Notes. — 1.  When  the  contents  and  two  dimensions  are  given,  the  other 
dimension  may  be  found  by  dividing  the  contents  by  the  product  of  the 
two  given  dimensions.     (Art.  119a,) 

2.  Excavations  and  embankments  are  estimated  by  the  cubic  yard.  In 
removing  earth,  a  cu.  yard  is  called  a  load. 

2.  How  many  cu.  feet  of  air  in  a  school-room  20  ft.  square 
and  lOi  ft.  high  ? 


180 


Comjpound  Numbers. 


3.  How  mauy  cu.  feet  in  a  mound  54  ft.  long,  36  ft.  wide, 
and  12  ft.  high  ? 

4.  How  many  loads  of  earth  must  be  removed  in  digging  a 
cellar  48  ft.  long,  25  ft.  wide,  and  8|-  ft.  deep  ? 

5.  What  will  it  cost  to  dig  such  a  cellar,  at  33-|  cts.  a  cu. 
yard  ? 

6.  What  will  it  cost  to  fill  in  a  street  55  feet  wide,  GOO  ft. 
long,  and  5|  ft.  below  grade,  at  42  cents  a  cu.  yard  ? 

7.  What  is  the  volume  of  a  cube  whose  edge  is  5  yd.  2  ft. 
6  in.  ? 

8.  Find  the  volume  of  a  cube  whose  edge  is  15 J  ft.  ? 

9.  The  width  of  a  reservoir  is  24  ft.,  its  depth  8  ft.,  and  its 
volume  5760  cu.  ft. ;  what  is  its  length  ? 


Wood   Measure,  , 

430.  A  Cord  of  Wood  is  a  pile  8  feet  long,  4  feet  wide,  and 
4  feet  high.     (Art.  353.) 

431.  A  Cord  Foot  is  1  foot  in  length  of  such  a  pile.    Hence, 

1  cord  foot  ^16  cubic  feet 
8  cord  feet  =  1  cord. 


1.  How  many  cords  of  wood  in  a  pile  35  ft.  long,  6  ft.  high, 
and  4  ft.  wide  ? 

2.  Find  the  number  of  cords  in  a  pile  of  wood  42  ft.  long, 


5-|  ft.  high,  and  8  ft.  wide. 


3.  At  $4.25  a  cord,  what  will  a  pile  of  wood  26  ft.  long,  4  ft. 
wide,  and  4  ft.  high  cost  ? 

4.  In  3  cord  feet,  how  many  cu.  feet  ?    In  5  cord  feet  ? 


Masonry.  181 

5.  How  many  cubic  feet  in  12  cords  ?     In  24  cords  ? 

6.  If. a  pile  of  wood  is  28  ft.  long^  4  ft.  wide,  how  high  must 
it  be  to  contain  112  cords  ? 

7.  How  many  cord  feet  in  a  load  of  wood  8  ft.  long,  4  ft. 
high,  and  3  ft.  wide  ? 

8.  What  is  the  worth  of  a  pile  of  wood  4  ft.  in  height,  6  ft. 
in  length,  and  o^  in  width,  at  $4.50  per  cord  ? 

9.  What  must  be  the  height  of  a  load  of  wood  that  is  6  ft. 
long  and  4  ft.  wide,  to  contain  a  cord  ? 


Masonry. 

432.  Stone  Masonry  is  sometimes  estimated  by  the  percli. 
Brickwork  is  estimated  by  the  thousand  bricks. 

Notes. — 1.  A  perch  of  stone  masonry  is  164  ft.  long,  li  ft.  wide,  and 
1  ft.  liigli,  which  is  equal  to  24J  cu.  ft.  It  is  customary,  however,  to  call 
25  cu.  ft.  a  perch. 

2.  The  average  size  of  bricks  is  8  in.  long,  4  in.  wide,  and  2  in.  thick. 

In  eBtimating  the  labor  of  brickwork  by  cu.  feet,  it  is  customary  to 
measure  the  length  of  each  wall  on  the  outside ;  no  allowance  being  made 
for  windows,  doors,  or  corners.  But  a  deduction  of  -^^  the  solid  contents 
is  made  for  the  mortar. 

1.  How  many  perch  (25  cu.  ft.)  in  the  walls  of  a  cellar,  the 
thickness  of  which  is  1  ft.  6  in.,  the  height  8  ft.,  each  side  wall 
being  42  ft.,  and  each  end  wall  24  feet  ? 

2.  At  $4.75  a  perch,  what  will  it  cost  to  build  the  walls  of 
the  above  cellar? 

3.  How  many  bricks  will  it  take  to  build  the  walls  of  a  house 
50  ft.  long,  25  ft.  wide,  21  ft.  high,  and  1  ft.  thick,  deducting 
-^  of  the  contents  for  the  mortar,  but  making  no  allowance  for 
windows  and  doors  ? 

4.  How  many  bricks  will  be  required  to  build  a  house,  the 
walls  of  which  are  48  ft.  long,  24  ft.  wide,  42  ft.  high,  and  1  ft. 
thick,  making  no  allowance  for  windows,  doors,  or  corners  ? 

5.  At  83.50  per  M.  for  bricks,  deducting  -^  for  mortar,  and 
$4.25  per  M.  for  laying  them,  what  will  the  walls  of  such  a  house 
cost? 


182  Compound  Nuinbers, 


Board    Measure. 

433.  A  Board  Foot  is  1  ft.  long,  1  ft.  wide,  and  1  in.  thick; 
that  is,  a  square  foot  1  inch  thick. 

434.  A  Board  Inch  is  ^^  of  a  board  foot ;  that  is,  1  inch 
long  by  12  inches  wide  and  1  inch  thick.  Hence,  Twelve 
board  feet  are  equal  to  1  cubic  foot. 

435.  Sawed  timber,  as  plank,  joists,  etc.,  is  estimated  by 
cu.  feet ;  heivn  timher,  as  beams,  etc.,  either  by  board  feet  or 
cu.  feet;  round  timber,  as  masts,  etc.,  by  cu.  feet. 

Written     Exercises. 

436.  To  find  the  Contents  of  Boards,  Planks,  etc. 

1.  How  many  board  feet  in  a  board  11  ft.  long,  18  in.  wide, 
and  1  inch  thick  ? 

Explanation.— Multiplying  the  length  operatiok. 

in  feet  by  tlie  width  and  thickness  expressed  11   X  18   X  1  =  198  in. 
in  inches,  we  have  198  board  inches.     Di-  198-^-12  =  16^  ft. 

viding  this  product  by  12,  the  result  is  16|  Ans     IQ^  ft 

board  feet,  Arts. 


2 


2.  How  many  board  feet  in  a  scantling  14  ft.  long,  4  in. 
wide,  and  2^  in.  thick  ? 

Solution. — Multiplying  the  length  in  feet  by  the  width  and  thickness 
expressed  in  inches,  we  have  14  ft.  x  4  x  2|^  =  140,  and  140  -f-  12  =  llf 
board  ft.,  Ans.    Hence,  the 

EuLE. — Multiply  the  length  in  feet  hy  the  width  and 
thichness  expressed  in  inches,  and  divide  the  product  hy 
12 ;  the  quotient  will  he  in  hoard  feet. 

Notes. — 1.  The  standard  thickness  of  a  board  is  1  inch.  If?^5sthan 
1  inch,  it  is  disregarded  ;  if  more  than  1  inch,  it  becomes  a  factor  in  find- 
ing the  contents  of  plank,  scantling,  etc. 

2.  If  aboard  is  ^a^enVi^,  multiply  the  length  by  half  the  sum  of  the 
two  ends. 


Board  Measure.  183 

3.  The  approximate  contents  of  round  timher  or  logs  may  be  found  by 
multiplying  \  of  the  mean  circumference  by  itself,  and  this  product  by  the 
length. 

3.  What  is  the  number  of  feet  in  a  board  14  ft.  long  and 
17  in.  wide  ? 

4.  Find  the  contents  of  a  tapering  board  15  ft.  long,  17  in. 
wide  at  one  end  and  11  in.  at  the  other  ? 

5.  Required  the  contents  of  8  boards,  11  ft.  long  and  15  in. 
wide  ? 

6.  What  is  the  worth  of  120  boards  of  the  above  size  at 
4  cents  a  board  foot  ? 

7.  Find  the  contents  of  a  board  16  ft.  long,  15  in.  Avide,  and 
J  in.  thick  ? 

8.  What  are  the  contents  of  8  scantlings  15  ft.  long,  4  in. 
wide,  and  3  in.  thick,  board  measure  ? 

9.  How  many  feet  in  a  beam  16  ft.  long,  8  in.  wide,  and 
4  in.  thick,  board  measure  ?     Cubic  measure  ? 

10.  What  cost  24  joists  whose  dimensions  are  4  in.  by  3  in. 
and  11  ft.  long,  at  25  cts.  a  cu.  foot  ? 

11.  Wbat  must  be  the  length  of  a  piece  of  timber  16  in.  by 
15  in.,  to  contain  20  cu.  feet  ? 

12.  How  many  cu.  feet  in  a  log  65  ft.  long,  whose  mean 
circumference  is  8  ft.  ? 

13.  How  many  cords  of  wood  in  such  a  log  ? 

14.  How  many  feet  of  inch  boards  will  it  take  to  build  a 
fence  4  ft.  high  and  125  ft.  long  ? 

15.  At  $2.25  per  100  ft.,  what  will  the  boards  cost  for  such 
a  fence  ? 

16.  What  amount  of  inch  boards  would  be  required  to  make 
a  box  4  ft.  long,  3 J  ft.  wide,  and  2J  ft.  deep  ? 

17.  AVhat  is  the  cost  of  a  stock  of  9  boards  14  ft.  long  15  in. 
wide,  at  $23.50  per  1000  ft. 

18.  How  many  cu.  feet  in  a  mast  54  ft.  long,  the  circumfer- 
ence of  which  is  9  ft. ;  and  what  will  it  cost  at  $1.09  a  cu.  foot  ? 

19.  What  cost  12  planks  14  ft.  long,  12|^  in.  wide,  and  2|-  in. 
thick  ;  at  $18  per  M.  ? 

20.  How  many  cu.  feet  in  a  log  62  ft.  long  and  28  ft.  in  cir- 
cumference ? 


i84  Compound  Numbers. 


Rectangular   Cisterns,  Bins,   Etc. 

437.  The  Capacity  of  rectangular  cisterns,  bins,  etc.,  is 
measured  by  cubic  measure,  but  tlie  results  are  commonly 
expressed  in  units  of  Liquid  and  Dry  Measure. 

438.  To  find  the  Number  of  Gallons  in  Rectangular  Cisterns,  etc. 

1.  How  many  gal.  of  water  will  a  rectangular  cistern  6  ft.  long, 
4  ft.  wide,  and  3  ft.  deep  contain  ? 

Analysis.— The  product  of  6  ft.  x  4  x  3  =  72  cu.  feet  in  the  cistern  ; 
and  73  x  1728  =  124416  cu.  inches.  Again,  in  1  gallon  there  are  231  cu, 
inches,  and  124416-T-231  :ir  538f?  gal.,  Ans.    (Ai-t.  357.) 

2.  How  many  bushels  in  a  bin  11  ft.  long,  4  ft.  wide,  and 
3  ft.  high  ? 

Analysis.— 11  fl.  x  4  x  3  =  132  cu.  feet,  and  132  cu.  ft.  x  1728  =  228096 
cu.  inches.  Now  1  bu.  contains  2150.4  cu.  in.  and  228096  cu.  in.  -^  2150.4 
=  106 i^j  bu.,  Ans.       (Art.  360.)     Hence,  the 

Rule. — Find  the  iiiunber  of  cubic  inches  in  the  thing 
measured,  and  reduce  them  to  liquid  or  dry  measure,  as 
may  he  requij^ed.     (Arts.  356,  359.) 

3.  Find  the  number  of  gallons  in  a  cistern  8  ft.  long  by  6  ft. 
wide  and  5  ft.  deep. 

4.  How  many  hogsheads  in  a  tank  12  ft.  square  and  8  feet 
deep  ? 

5.  In  a  reservoir  40  ft.  long,  30  ft.  wide,  and  15  ft.  high, 
how  many  hogsheads  ? 

6.  I  wish  to  build  a  cistern  containing  5000  gal.,  whose  base 
is  12  ft.  by  8 ;  what  must  be  its  height  ? 

7.  If  a  reservoir  45  ft.  long,  28  ft.  wide,  contains  40000  hhd., 
how  high  must  it  be  ? 

8.  At  $1.12J  a  bushel,  what  is  the  value  of  a  bin  of  wheat 
9  ft.  long,  5  ft.  wide,  and  4  ft.  deep  ? 

9.  How  many  cu.  feet  in  a  bin  which  will  contain  300 
bushels  of  grain  ? 


Mectangidar  Cisterns,  Bins,  JEtc.  i85 

.  439.  Shorter  Methods.— Since  2150.4  cu.  inches  -r-  1728  cu. 
in.  =  1:1,  it  follows  that  a  bushel  must  contain  1^  cu.  ft. 
nearly.     (x\rt.  360.)     Hence,  we  have  the  following  methods : 

1st.  Divide  the  number  of  cu.  feet  in  a  bin  by  1^  and  the 
quotient  will  be  the  approximate  number  of  bu.  in  the  bin. 

2d.  Multiply  the  number  of  bu.  in  a  bin  by  1^,  and  the  pro- 
duct will  be  the  approximate  number  of  cu.  feet  in  the  bin. 

3rd.  A  ton  (2000  lbs.)  of  Lehigh  white  ash,  ^gg  size,  coal 
in  bins  measures  34|^  cu.  ft. 

A  ton  of  white  ash  Schuylkill,  Qgg  size,  measures  35  cu.  ft. 
A  ton  of  pink,  gray,  and  red  ash,  egg  size,  measures  3G  cu.  ft. 

4th.  A  ton  of  hay  upon  a  scaffold  measures  about  500  cu.  ft. ; 
when  in  a  mow,  400  cu.  feet ;  and  in  well  settled  stacks, 
10  cubic  yards. 

10.  How  many  bushels  of  corn  can  be  put  into  a  bin  6  ft. 
long,  5  ft.  wide,  and  4  ft.  deep  ?     A7is.  96  bushels. 

11.  A  farmer  has  a  bin  10  ft.  long,  GJ  ft.  wide,  and  4  ft. 
deep  ;  how  many  bushels  does  it  hold  ? 

12.  A  bin  holding  150  bu.  is  6  ft.  wide  and  4  ft.  deep  ;  what 
is  its  length  ? 

13.  A  bin  containing  280  bushels  is  10  ft.  long  and  7  ft. 
wide ;  what  is  its  depth  ? 

14.  "What  must  be  the  length  of  a  bin  8  ft.  wide,  5  ft.  deep, 
to  contain  320  bushels  ? 

15.  At  ll-J  a  bushel,  what  is  the  value  of  a  bin  of  wheat  12.5 
ft.  long,  6  ft.  wide,  and  4  ft.  deep  ? 

16.  A  farmer  filled  a  bin  8  ft.  long,  7  ft.  wide,  and  5  ft. 
deep,  with  the  corn  raised  on  5  acres ;  how  many  bushels  was 
that  per  acre  ? 

17.  How  many  tons  of  Lehigh  white  ash,  egg  size  coal,  "g^ill 
fill  a  bin  12  ft.  long,  8  ft.  wide,^  6  ft.  high  ? 

18.  How  many  tons  of  hay  in  a  mow  20  ft.  long,  18  ft.  wide, 
and  14  ft.  high  ? 


186  Compound  Ntiynhers. 

Oral    Problems    for    Review, 

440.    1.  At  3  cts.  a  yd.,  what  will  5  mi.  of  telegraph  wire  cost? 

2.  My  neighbor's  farm  is  f  mile  square  ;  how  many  acres  did 
it  contain  ? 

3.  Bought  40  acres  of  land  at  75  cts.  per  sq.  rod,  and  sold  it 
so  as  to  double  my  money  ;  required  my  gain  ? 

4.  At   25  cts.    a   gallon,    what  is   a  family's   milk  bill  for 
60  days,  taking  2  qts.  daily  ? 

5.  If  a  man  lives  2|  miles  from  the  City  Hall,  how  many 
miles  will  he  travel  in  6  days,  making  1  trip  a  day  ? 

6.  The  length  of  a  blackboard  is  6  ft.,  its  width  4  ft. ;  how 
many  sq.  yards  does  it  contain  ? 

7.  If  a  garden  is  5  rods  long  and  4  rods  wide,  how  many 
rods  in  its  perimeter  ? 

8.  If  1  oz.  of  spice  costs  8  cents,  what  will  2J  pounds  cost  ? 

9.  At  II  a  sq=  yard,  what  will  it  cost  to  carpet  a  room 
18  feet  long  and  15  ft.  wide  ? 

10.  A  stationer  paid  11.25  a  gross  for  pencils,  and  sold  them 
for  a  cent  apiece  ;  how  much  did  he  gain  on  5  gross  ? 

11.  In  a  certain  school  are  72  girls,  and  f  of  the  pupils  are 
boys;  how  many  pupils  in  the  school  ? 

12.  The  surface  of  a  cube  is  150  sq.  inches ;  what  is  the  sur- 
face of  one  side  ? 

13.  What  fraction  of  a  semi-circumference  is  45  degrees  ? 

14.  How  many  writing  books  of  36  pages  each  can  be  made 
from  a  half  ream  of  pajoer  ? 

15.  How  many  days  in  7  of  the  longest  months  ? 

16.  A  can  do  a  job  in  2  days,  B  in  3  days ;  what  part  will 
each  do  in  one  day  ? 

17.  How  long  will  it  take  both  to  do  the  same  job  working 
together  ? 

18.  How  many  yards  of  carpeting  j  yd.  wide,  will  carpet  a 
room  18  ft.  square  ? 

19.  How  many  sq.  yards  in  the  pavement  of  a  street  60  ft. 
wide  and  800  ft.  long  ? 

20.  How  many  suits  of  clothes  can  be  made  from  648  yards, 
allowing  4  yds.  to  a  suit  ? 


Review,  187 


Written    Problems    for    Review. 

441.     1.  How  many  acres  in  a  piece  of  land  18i  rods  long  and 
96  rods  wide  ? 

2.  What  will  16568  cu.  feet  of  wood  cost,  at  13^  a  cord  ? 

3.  How  many  dollars  can  be  made  out  of  50  lb.   9  oz.  of 
silver,  allowing  ^Vl\  grains  to  a  dollar  ? 

4.  How  many   cubic   inches   in   a  box  whose  length  is  30 
inches,  its  breadth  18,  and  its  depth  15  inches  ? 

5.  How  many  cubic  inches  in  a  block  of  marble  43  inches 
long,  18  inches  broad,  and  12  inches  thick  ? 

6.  How  many  cubic  feet  of  air  in  a  school-room  16  feet  long, 
15  feet  wide,  and  9  feet  high  ? 

7.  How  many  cubic  feet  in  a  pile  of  wood  16  feet  long,  6 
feet  wide,  and  5  feet  high  ?     How  many  cords  ? 

y        8.  How  many  cords  of  wood  in  a  pile  140  feet  long,  4|-  feet 
wide,  and  ^\  feet  high  ? 

9.  At  50  cts.  per  decister,  what  will  a  ster  of  wood  cost  ? 

10.  What  will  a  metric  ton  of  h^mp  cost,  at  25  cts.  per  kilo? 

11.  At  6  cts.  per  liter,  what  cost  a  hektoliter  of  milk? 

X      12.  How  many  square  yards  in  the  four  sides  of  a  room  18 

feet  long,  Vl\  feet  wide,  and  14J  feet  high  ? 
N         13.  How  many  square  yards  of  plastering  will  it  take  to 
cover  the  four  sides  and  the  ceiling  of  a  room  18  feet  square, 
and  15  feet  high  ? 

14.  How  many  yards  of  muslin  3  qrs.  wide,  are  equal  to  36 
yds.  brocatelle,  which  is  \\  yard  wide  ? 

15.  How  many  yards  of    silk  3  qrs.  wide,  will  51  yds.  of 
cambric  line,  which  is  IJ  yd.  wide  ? 

16.  What  will  it  cost  to  pave  a  street  3  mi.  115  rods  long, 
and  2  rods  wide,  at  %\h\  a  square  rod  ? 

17.  A  man  having  15  acres  and  60  rods  of  land,  laid  it  out 
^  in  lots  each  containing  12  sq.  rods,  and  sold  the  lots  at  $150 

apiece ;  how  much  did  he  realize  for  his  land  ? 

18.  What  is  the  worth  of  a  pile  of  wood  18  ft.  long,  10-J  ft. 
high,  and  9J  wide,  at  %'6\  a  cord  ? 

19.  How  many  times  will  a  wheel  of  a  railroad  car,  9  ft.  in 
circumference,  turn  round  in  going  1500  miles  ? 


188  Compound  Numbers. 

20.  How  long  would  it  take  a  cannon  ball,  flying  at  the 
rate  of  8  miles  per  minute,  to  reach  the  moon,  a  distance  of 
240000  miles  ? 

21.  The  velocity  of  light  is  11875000  miles  per  minute,  and 
it  takes  8  minutes  for  it  to  pass  from  the  sun  to  the  earth ;  how 
far  from  the  sun  is  the  earth  ;  and  how  many  weeks  would  it 
take  to  travel  this  distance,  30  miles  an  hour  ? 

^yC^  22.  How  many  bricks  will  it  take  to  pave  a  sidewalk  75  feet 
long  and  8  feet  wide,  each  brick  being  8  inches  long  and 
4  inches  wide  ? 

23.  Required  to  reduce  5  mi.  6  fur.  23  rods  5  yd.  and  8  in. 
to  inches,  and  prove  the  operation. 
Y    24.  Allowing  1  shingle  to  cover  24  sq.  inches,  how  many 
shingles  will  be  required  to  cover  the  roof  of  a  house  50  feet 
long,  the  rafters  on  each  side  being  29  feet  long  ? 

25.  How  many  farms  of  160  A.  in  a  township  6  miles  square  ? 

26.  How  many  bricks  will  it  take  to  build  a  prison  60  feet 
long,  25  feet  wide,  and  48  feet  high,  whose  walls  are  1  foot 
thick,  the  bricks  8  in.  long,, 4  in.  wide,  and  2  in.  thick  ? 

27.  If  the  pendulum  of  a  clock  vibrates  65  times  per  minute, 
how  much  time  will  it  gain  in  a  common  year  ? 

28.  How  many  years  would  it  take  to  count  a  billion,  count- 
ing 60  a  minute,  working  10  hours  a  day,  and  allowing  365 
days  to  a  year  ? 

Questions. 

418.  What  is  a  plane  figure  ?  419.  The  perimeter  of  a  plane  figure  ? 
430.  The  area  ? 

421.  What  is  a  rectangle  ?  428.  The  dimensions  of  a  rectangular  fig- 
ure ?  424.  How  find  the  area  of  rectangular  surfaces  ?  When  the  area 
and  one  side  are  given,  how  find  the  other? 

425.  What  is  a  rectangular  solid?  427.  The  contents  ?  428.  The  dimen- 
sions ? 

430.  What  are  the  dimensions  of  a  cord  of  wood?  431.  How  many 
cubic  feet  does  it  contain  ?     Cord  feet  ? 

432.  How  is  stone  masonry  estimated?    Brickwork  ? 

483.  What  is  a  board  foot  ?  434.  A  board  inch  ?  How  many  board  feet 
in  a  cu.  foot  ?  435.  How  are  sawed  and  liewn  timber  estimated  ?  Round 
timber?    436.  How  find  the  contents  of  boards,  plank,  etc.  ? 

438.  How  find  the  contents  of  cubical  bins,  cisterns,  etc.  ? 


EKCENTAGE. 


Oral     Exercises. 

442.  1.  When  a  number  is  divided  into  a  hundred  equ&! 
parts,  what  is  one  of  tlie  parts  called?  Two  of  the  parts? 
Five  ?     Ten  ? 

2.  A  man  paid  $100  for  a  horse  and  sold  it  for  $105 ;  how 
many  dollars  did  he  gain  ?  How  many  hundredths  of  the  cost 
did  he  gain  ? 

3.  What  part  of  $100  is  $5  ?     (Art.  226.) 

4.  If  I  pay  $100  for  a  sofa  and  sell  it  for  $94,  how  many 
dollars  shall  I  lose  ?    How  many  hundredths  of  the  cost? 

443.  The  number  of  hundredths  gained  or  lost  is  called 
the  Rate  j^er  cent. 

444.  Per  Cent,  means  by  the  hundred,  or  simply  hun- 
dredths. 

Thus,  8  per  cent  is  3  hundredtlis  of  a  number ;  5  per  cent  is  5  hun- 
dredths, etc. 

445.  The  Sign  of  Per  Cent  is  %.  Thus,  4:%  means  4  per 
cent. 

446.  The  process  of  calculating  by  hundredths  is  called 
Percentage. 

5.  A  farmer  lost  8  sheep  out  of  every  100  of  his  flock  ;  what 
per  cent  of  them  did  he  lose  ? 

6.  A  man  gave  away  810  out  of  every  $100  of  his  income; 
what  per  cent  of  his  income  did  he  give  away  ? 

7.  A  teacher  having  a  class  of  150  pupils,  promoted  10^  of 
them  ;  how  many  were  promoted  ? 


190  Percentage. 

447.  Per  cent  is  expressed  by  decimals,  by  %,  or  by  fractions. 


Tab  le 


Sign. 

Decimal. 

Fraction. 

Sign. 

Decimal, 

Fraction 

H 

.01 

—          100 

\% 

.005 

= 

200 

H 

.05 

—  1 

—  20 

H% 

.035 

= 

1 
"20¥ 

10^ 

.10 

—  1 

—  10 

i% 

.0025 

=^ 

40  0 

25%^ 

.25 

i 

H% 

.0625 

= 

1 

50^ 

.50 

—                 1 
2 

m% 

.1875 

zzz 

A 

75^ 

.75 

t 

3H% 

.33i 

— 

i 

100^ 

1.00 

—  i 

—  1 

mi% 

1.125 

^; 

H 

448.  Since  liwidredtlis  occupy  two  decimal  places,  every 
per  cent  requires,  at  least,  two  decimal  figures.  Hence,  if  the 
given  per  cent  is  less  than  10,  a  cipher  must  be  prefixed  to  the 
figure  denoting  it.     Thus,  2^  is  written  .02;  6^,  .06,  etc. 

Notes. — 1.  A  hundred  per  cent  of  a  number  is  equal  to  the  number 
itself;  for  {^^  is  equal  to  1. 

2.  In  expressing  per  cent,  when  the  decimal  point  is  used,  the  words 
per  cent  and  the  sign  {%)  must  be  omitted,  and  mce  versa.    Thus,  .05  de- 


but .05  per  cent  or  .05% 


notes  5  per  cent,  and  is  equal  to  jf  ^  or  ^ 
denotes  ^f  o  of  j^o.  and  is  equal  to  y^f^yo  or  ^oVo- 


449.  To  read  any  given  Per  Cent,  expressed  Decimally. 

Call  the  first  two  decimal  figures  per  cent ;  and  those  on  the 
right, ydecimal  parts  of  1  per  cent. 

Note. — Parts  of  1  per  cent,  when  easily  reduced  to  a  common  fraction, 
are  often  read  as  such.  Thus,  .105  is  read  10  and  a  half  per  cent ;  .0125  is 
iread  one  and  a  quarter  per  cent. 

Eead  the  following  as  rates  per  cent: 

8.     .06;  .052;  .085;  .094. 
.012;  .174;  .0836;  .154. 
.1857;  .2352;  .1685;. 7225. 


9. 
10, 
11. 

12.  1.07;  2.53;  4.65;  2.338. 

13.  5.33^;  4.125;  8.0623;  6.73f 


.12i;  .08i; 


.161;  .5775. 


Percentage,  191 

450.  Express  the  following  by  Com.  Frac.  in  lowest  terms  : 

14.  4:%.  17.     20^.  20.     75^.  23.     150^. 

15.  Q%.  18.     25^.  21.     100^.  24.     200^. 

16.  10^.  19.      50^.  22.      125^.  25.      500^. 

26.  To  what  common  fraction  is  8J-^^  equal  ? 

ANALYSis.-8i%  =  ^^i  ,  or  8i  -H  100  ;  and  8i  -=-  100  =  ^^  ^  ^  =  _2J^^. 
or  jV,  ^^«-     (-^rt.  220.) 

27.  To  what  common  fraction  is  ^%  equal  ? 
Analysis.— i^%  =  .005;  and  .005  =  yofo>  or  ¥¥o>  ^^'''-    (^t.  185.) 

28.  ^%  =  what  fraction  ?     37i^  ?     18|^  ?     2f %  ? 

Mental    Exercises. 

451.  1.  What  per  cent  of  a  number  is  -J-  of  it  ? 

Analysis. — Since  any  number  equals  100%   of  itself,  i  of  a  number 
must  equal  i  of  100%,  or  SS^^i,  Ans. 

2.  What  per  cent  of  a  number  is|-?     Ts|?    f?     |? 

3.  What  per  cent  of  a  number  isf?    J?    f?    f? 

4.  What  per  cent  of  a  number  is^?     ^''g-?    ^?     f? 

Written     Exercises. 

452.  To  change  a  Common  Fraction  to  an  equivalent  per  cent. 

1.  What  per  cent  of  a  number  is  -§-§-  of  it  ?  • 

Analysis. — Every  number  is  equal  to  100  %  operation. 

of  itself;  hence,  f^  of  a  number  =  f^  of  100%,  |A  =  24  -r-  60 

or  eV  of  2400%.     Therefore,  annexing  ciphers  QQ  )  24.00  (  .40,   A71S. 

to  the  numerator  and  dividing  by  the  denom-  oaq 

inator,  we  have  .40  or  40%.     Hence,  the  

KuLE. — A7vne.v  ciphers  to  the  numerator,  and  divide  it 
by  the  denominator.     (Art.  249.) 

2.  What  per  cent  of  a  number  is  ff  of  it  ? 
Ans.  .625,  or  62-1-;;. 

3.  What  per  cent  of  a  number  is  |4  oJ^  i^  ^     Is  -|f  ? 


192  Percentage, 

4.  What  per  cent  of  a  number  is  -^-^  of  it  ?    Is  ^  ? 

5.  What  per  cent  of  a  number  is  ff  ?     Is  -^^q-? 

6.  What  per  cent  of  a  number  is  ^^o  of  it  ?     Is  f  f  f  ? 

453.  The  Parti  or  Elements  employed  in  calculating  per- 
centage are  the  Base,  the  Rate  per  cent,  the  Percentage,  and 
the  Amount  or  Difference. 

454.  The  Base  is  the  number  on  which  the  percentage  is 
calculated. 

455.  The  Rate  is  the  number  of  Imndredtlis  of  the  Mse 
taken. 

456.  The  Percentage  is  the  part  of  the  base  indicated  by  the 
rate  per  cent. 

Thus,  when  it  ia  said  that  4%  of  $50  is  $2,  the  rate  is  .04,  the  base  $50, 
and  the  percentage  $2. 

457.  Tlie  Amount  is  the  stim  of  the  base  and  percentage. 

458.  The  Difference  is  the  base  less  the  percentage. 

Thus,  if  the  base  is  $75  and  the  percentage  $4,  the  amount  is  $75  +  4 
=  $79  ;  the  difference  is  $75- $4  =  $71. 

The  relation  between  these  parts  is  such,  that  if  any  two  of 
them  are  given,  the  other  three  may  be  found. 

PROBLEM     I. 

Oral     Exercises. 

459.  1.  What  is  h%  of  $60  ? 

Analysis. — 5%  of  a  number  equals  y^^,  or  ^^  of  the  number,  and  4x 
of  $G0  is  $3.     Therefore,  ^%  of  $60  is  $3. 


How  much  is 

How  much  is 

2.     4^  of  $80  ? 

7.     121%  of  320  rods  ? 

3.     Q%  of  1100  ? 

8.     20^  of  275  gallons? 

4.     1%  of  1200  ? 

9.     25^  of  260  acres  ? 

5.     8^  of  $400  ? 

10.     50%'  of  700  men  ? 

6.     10%  of  $250  ? 

11.     100/^  of  $2000  ? 

Percentage,  193 

12.  A  teacher  who  received  $30  a  month,  had  her  salary 
increased  V)%  ;  what  was  the  increase  per  month  ? 

13.  From  a  cistern  of  water  holding  a  hogshead  Z?>\%  leaked 
out  \  how  many  gallons  remained  ? 

Written     Exercises. 
460.  To  find  the  Percentage  when  the  Base  and  Rate  are  given. 

1.  What  is  9^  of  13465  ? 

OPKEATION. 

Analysis. — 9%  of  a  number  equals  yf^,  or  .09  of  it ;  $3465  B. 

therefore,  the  percentage  must  be  .09  times  $3465,  whicli  qq  -p 

is  equal  to  $311.85,  Ann.    Hence,  the  ' — 

$311.85  P. 

EuLE. — Multiply  the  hase  by  the  i^ate,  expressed  in 
decimals. 

Formula. — Percentage  =  Base  x  Rate. 

Notes. — 1.  When  the  rate  is  an  aliquot  part  of  100,  i\ie  percentage  may- 
be found  by  taking  a  like  part  of  the  base.  (Art.  447.)  Thus,  for  20%, 
take  i  ;  for  35  % ,  take  \,  etc. 

2.  When  the  hase  is  a  compound  number,  the  lower  denominations 
should  be  reduced  to  a  decimal  of  the  highest ;  or  the  whole  number  to 
the  lowest  denomination  mentioned  ;  then  apply  the  rule.     (Ex.  5.) 

3.  Finding  a  per  cent  of  a  number  is  the  same  as  finding  z.  fractional 
part  of  it.    (Art.  226.) 

2.  What  is  31%  of  1546  pounds  8  ounces  ? 
A71S.  572.205  pounds. 

Find  the  percentage  of  the  following  : 

3.  25^  of  $5068.  9.  50^  of  £2436. 

4.  42^  of  £6248.  10.  12^^  of  $2874. 

5.  75^  of  8675  bu.  3  pk.  ii.  22^5^  of  865  acres. 

6.  100^  of  2240  pounds.  12.  4:2^%  of  84820. 

7.  61^  of  $1000.  13.  62^^  of  4360  feet  6  in. 

8.  371^  of  $1568.  14.  33^^  of  $564175. 

15.  Which  is  greater,  7  per  cent  of  $6300,  or  6  p^r  cent 
of  $7200  ? 

9 


194  Percentage, 

16.  Which  is  less,  9  per  cent  of  182000,  or  6  per  cent  of 
$93000  ? 

17.  A  man  had  18750  in  bank  and  drew  out  8^  of  it  at  one 
time,  and  then  10^  of  the  remainder ;  how  much  had  he  left 
on  deposit  ? 

18.  A  man  who  owed  19584  failed  in  business  and  paid  40^ 
of  his  debts  ;  how  much  did  he  pay  ? 

19.  A  land  speculator  paid  $6075  for  a  farm,  and  sold  it  at 
\h%  less  than  cost ;  how  much  did  he  lose  ? 

461.  The  Amount  is  found  by  adding  the  percentage  to  the 
base. 

462.  The  Difference  by  subtradmg  the  percentage  from 
the  base. 

,-,  j  Amomit     =  Base  +  Percentage. 

I  Difference  =  Base  —  Percentage. 

20.  A  began  business  with  $4200  capital,  and  increased  it 
7  per  cent  the  first  year ;  what  amount  of  caj^ital  did  he 
then  have  ? 

Solution.— $4200  X  .07  =  $394.00,  and  $4200 +  $294  =  $4494,  Ans. 

21.  B  commenced  business  with  16500  capital,  and  lost 
6  per  cent  of  it  the  first  year ;  how  much  capital  had 
he  then  ? 

Solution.— $6500  x  .06  =  $390.00,  and  $6500- $390  :=  $6110,  Ans. 

22.  A  man  sold  his  house,  which  cost  liim  $5760,  at  12^^ 
above  cost ;  what  amount  did  he  receive  for  his  house  ? 

23.  A  farmer  raised  4256  HI.  of  grain,  and  sold  12^^^'  of  it ; 
how  many  hektoliters  did  he  have  left  ? 

24.  What  is  the  amount  of  $252500  increased  by  20^  of 
itself  ? 

25.  A  commander  having  an  army  of  16293  men,  lost  33^^^ 
of  them  by  sickness  and  desertion ;  how  many  soldiers 
remained  ? 

26.  A  farmer  owning  3500  sheep,  lost  50  per  cent  of  them 
by  disease  ;  how  many  had  lie  left  ? 

27.  If  my  annual  income  is  13560,  and  I  spend  25^  of  it 
each  year,  how  much  shall  I  save  in  4  years  ? 


Percentage,  195 


PROBLEM     II. 

Oral     Exercises. 

463.  1.  A  farmer  had  100  sheep  and  lost  50  of  them ; 
what  part  of  them  did  he  lose  ?  How  many  hundredths  ? 
How  many  per  cent  ? 

Analysis. — 50  is  equal  to  -^§i^,  or  1  half ;  and  since  per  cent  means 
hundredths,  -^^^  equals  50  per  cent. 

2.  A  man  spent  $25  for  a  suit  of  clothes,  which  was  |-  of  his 
money  ;  what  per  cent  of  his  money  did  he  spend  ? 

3.  A  pupil  missed  \  of  his  questions;  how^  many  hundredths 
did  he  miss?     How  many  per  cent  ? 

4.  What  part  of  $12  is  13  ?     What  per  cent  ? 

5.  What  per  cent  of  20  is  7  ? 

Analysis.— 7  is  ^  of  20 ;  and  20  is  100;^  of  itself.  Now  J^  of  100 /o  is 
5%,  and  ^^  of  100;:^  is  7  times  5,  or  35^. 

What  per  cent  What  per  cent 

6.  Of  $25  are  $8  ?  13.   Of  $24  are  $18  ? 

7.  Of  $10  are  S9  ?  14.   Of  20  pears  are  12  pears  ? 

8.  Of  5  is  3  ?  15.   Of  25  gal.  are  16  gal.  ? 

9.  Of  16  is  4  ?  16.   Of  50  lb.  are  45  lb.  ? 

10.  Of  36  is  9  ?  17.   Of  $50  are  $12i  ? 

11.  Of  63  is  31i  ?  18.   Of  $1  are  6^  cents  ? 

12.  Of  16f  is  81  ?  19.   Of  $1  are  33i  cents  ? 

20.  If  you  pay  $5  for  the  use  of  $50  for  a  year,  what  per 
cent  do  you  pay  ? 

21.  What  per  cent  of  30  kilograms  are  6  kilograms  ? 

22.  If  a  pint  of  water  is  added  to  a  gallon  of  milk,  what  per 
cent  of  it  is  water  ? 

23.  If  a  man  earns  $80  a  month  and  spends  $30,  what  per 
cent  does  he  spend? 


196  Percentage. 

Written     Exercises. 
464.  To  find  the  Hate  when  the  Base  and  Percentage  are  given. 

1.  What  per  cent  of  $63  is  142  ? 

Analysis.— Percentage  is  the  product  of  the  hase  operation. 

and  the  rate  ;  therefore,  the  percentage  $42,  divided  by  63  )  42.00 

the  base  S63,  gives  .66 1,  or  60 1 ;^,  the  rate.     Hence,  the  Ans,  .^^^ 

Rule. — Divide  the  percentage  hy  the  hase. 

Formula. — Rate  =  Percentage  -^  Base. 

2.  What  %  of  £18  is  15s.     Ans.  4^%- 

3.  What  %  of  96  meters  is  28  meters  ? 

4.  What  %  of  $18  is  12  cts.  ? 

5.  iVhat  %  of  168  is  15  ? 

6.  What  %  of  275  is  18  ? 

7.  What  %  of  15  is  5}  dimes  ? 

8.  What  %  of  4  ton  is  ^  ton  and  16  ponnds  ? 

9.  Henr}^  spelled  225  words  out  of  250,  and  his  sister  235  ; 
what  per  cent  of  the  words  did  each  spell  correctly  ? 

10.  From  a  cask  of  kerosene  containing  52  gal.,  6  gal.  2  qts. 
leaked  ont;  what  per  cent  of  it  was  lost  ? 

11.  A  farmer  haying  250  bu.  of  wheat,  sold  |  of  it ;  how 
many  bnshels  and  what  per  cent  did  he  sell  ? 

12.  A  man  worth  $12500,  beqneathed  $3125  to  bis  wife  and 
the  rest  to  his  3  children  ;  what  per  cent  of  it  did  his  wife 
have,  and  how  much  had  each  child  ? 

13.  What  per  cent  of  365  days  are  30  days  ? 

14.  Of  1880  years  are  4000  years  ? 

15.  Of  27  lb.  Avoir,  are  12  oz.  ? 

16.  Of  125  miles  are  250  rods  ? 

17.  Of  88  kilograms  are- 75  grams  ? 

18.  If  a  man  owns  f  of  a  ship  and  sells  |  of  her,  what  per 
cent  of  his  part  does  he  sell  ? 

19.  What  per  cent  of  75  bu.  3  pk.  are  50  bu.  2  pk.  ? 

20.  A  man  gave  19863  to  3  charities ;  to  the  first  $2500,  to 
the  second  $4500 ;  how  much  was  left  for  the  third  and  what 
per  cent  did  each  receive  ? 


Percentage,  197 


PROBLEM     III. 

Oral     Exercises. 

465.  1.  $48  are  %%  of  what  number  ? 

Analysis. — Since  $48  are  6  5^  of  the  number,  1%  is  ^  of  $48,  which  is 
8,  and  100^  is  100  times  8,  or  $800,  Am. 

2.  24  is  4^  of  what  ?  7.  40  gal.  are  20^  of  what  ? 

3.  32  is  h%  of  what  ?  8.  $68  are  12^  of  what  ? 

4.  48  is  20g  of  what  ?  9.  25  yd.  are  40%'  of  what  ? 

5.  12^^  is  10^  of  what  ?  10.  12d.  are  30^  of  what  ? 

6.  6}  is  25^  of  what  ?  11.  25  doz.  are  12^^  of  what  ? 

Written     Exercises. 

466.  To  find  the  Base  when  the  Rate  and  Percentage  are  given. 

1.  192  is  25^  of  wliat  number  ? 

Analysis. — Percentage  is  the  product  of  the  .25  )  192.00     P. 

base  by  the  rate.     The  base  192-^.25  =  768,  the  . — T 

base  required.     Hence,  the  ^^^*'-  "^^     ^' 

Rule. — Divide  the  percentage  hy  the  rate. 

Formula. — Base  =  Percentage  -^  Rate. 

2.  84  is  12|-/^^  of  what  number?       Ans.  672.     (Art.  447.) 

3.  96  =  ^d\%  of  Avhat  ?  9.  31.25  =  12^%  of  what  ? 

4.  234  =  10^  of  what  ?  10.  60  cts.  =  1%  of  what  ? 

5.  £240  =  7%  of  what  ?  11.  $100  =  1%  of  what  ? 

6.  62.5  =  61^  of  what  ?  12.  $42.30  =  \%  of  what  ? 

7.  60  yd.  =  1%  of  what  ?  13.  94  =  Vd{)%  of  what  ? 

8.  78  =  25;:^  of  what  ?  14.  58^  =  125%  of  what  ? 

15.  The  number  of  children  of  age  to  attend  school  is  862, 
which  is  20;^  of  the  population  ;  what  is  the  whole  population  ? 

16.  A  man  sold  a  house,  making  $360,  which  was  b%  more 
than  it  cost  him  ;  what  did  he  pay  for  the  house  ? 

17.  4.%  of  $230  is  b%  of  what  ?  "^  12|-/^^'  of  $530  is  6^%  of  what  ? 

18.  A  man  paid  a  war  tax  of  $73.50,  which  was  2%  on  the 
value  of  his  property  ;  what  was  he  worth  ? 


198  Percentage, 


PROBLEM     IV. 

Oral     Exercises. 

467.  1.  A  man  sold  a  cow  for  140,  which  was  25^  more 
than  she  cost  him ;  what  did  he  pay  for  her  ? 

Analysis. — $40  is  the  cost  increased  by  25%  of  itself  :  and  since  the 
cost  is  \%%  of  itself,  $40  must  be  \%%,  or  f  of  the  cost.  Now,  as  $40  —  |  of 
the  cost,  \  is  \  of  $40,  which  is  $8,  and  f  are  4  times  8,  or  $32,  Ans. 

2.  What  number  increased  by  25^  of  itself,  is  100  ? 

3.  A  furniture  dealer  sold  a  bureau  for  $20,  which  was  10^ 
more  than  it  cost  him ;  how  much  did  it  cost  him  ? 

4.  What  number  plus  12^^  of  itself  amounts  to  96  ? 

.  A  grocer  sold  a  barrel  of  apples  for  $5.50,  and  gained  20^ 
on  the  sum  it  cost  him  ;  what  did  he  pay  for  it  ? 

6.  A  jeweller  sold  a  watch  for  $150,  which  was  50^  more 
than  it  cost  him  ;  what  did  he  pay  for  it  ? 

7.  What  number  diminished  by  25^  of  itself  is  60  ? 
Analysis. — As  60  is  the  number  after  it  is  diminished,   60  must  be 

100%  —25fo  =  iVo>  or  f  of  the  number.     Now  if  60  is  f  of  the  number,  ^ 
of  it  is  60-^3  =  20,  and  4  fourths  are  4  times  20,  or  80,  Ans. 

8.  What  number  diminished  by  20;^  of  itself  is  48  ? 

9.  A  pupil  answered  on  examination  45  questions  correctly, 
which  was  10^  less  than  the  number  asked  him ;  how  many 
were  asked  him,  and  how  many  did  he  miss  ? 

Written     Exercises. 

468.  To  find  the  Base  when  the  Amount  or  Difference,  and  the 

Rate  are  given. 

1.  What  number  increased  by  25^  of  itself  is  3500  ? 
Analysis. — Since  3500  is  the  number  after  it  is        1  -j-  .25  =  1.25 

increased  by  25%  of  itself,  3500  must  be  125%  of  1.25  )  3500.00 

the  number,  or  1.25  times  the  number,  and  3500  — 

-125  =  2800,  Ans.  ^^^-    ^^00 

2.  What  number  diminished  by  20^  of  itself  is  2560  ? 

Analysis.— Since  2560  is  the  number  after  it  is  ■'•        .-vO  =:  .80 

diminished  by  20%  of  itself,  2560  must  be  80%,  or  .80  )  2560.00 

.80  times  the  number,  and  2560-T-.80  =  3200,  Ans.  Ans    3200 


Percentage,  199 

469.  From- the  operations  above,  we  derive  the  following 

Rule. — Divide  the  amount  hy  1  increased  hy  the  rate. 
Or,  Divide  the  difference  hy  1  diminished  hy  the  rate, 

^  „  (  Amount  -^  (1  +  Rate), 

Formulas. — Base  =  \  ^..^,  '    .        ^  ,  . 

(  Difference  -^  (1  —  Rate). 

What  number  increased  What  number  diminished 

3.  By  !()%  of  itself  is  5342  ?      9.  By  25%  of  itself  is  3900  ? 

4.  By  %%  of  itself =2418  ?       10.  By  Q%  of  itself =2100  ? 

5.  By  105^  of  itself =28600  ?    ii.  By  12%  of  itself =1200  ? 

6.  By  16%  of  itself =2552  ?      12.  By  15%  of  itself =2300  bu.? 

7.  By  20%  of  itself  =  .$3720?    13.  By  7|%  of  itself =$6475  ? 

8.  By  28i%  of  itself =18995  ?  14.  By  12^%  of  itself =13125  ? 

15.  At  thQ  end  of  the  year,  a  merchant's  stock  was  18400, 
which  was  17%  more  than  his  capital ;  what  was  his  capital? 

16.  A  man  sold  his  house  for  $2700  and  lost  12  J  %  ;  what  did 
the  house  cost  him  ? 

17.  A  grocer  sold  950  barrels  of  flour  for  $5760,  which  was 
20%  advance  on  the  cost ;  Avhat  was  the  entire  cost,  and  the 
cost  per  barrel  ? 

18.  A  provision  dealer  sold  800  barrels  of  beef  for  112000, 
which  was  a  loss  of  25% ;  what  Avas  the  whole  cost,  and  how 
much  per  barrel  ? 

-  470.  Percentage  is  applied  to  two  classes  of  problems. 

First. — Those  which  are  independent  of  Time  ;  as.  Profit  and 
Loss,  Commission  and  Brokerage,  Insurance,  Taxes,  Duties. 

Second. — Tliose  in  which  Time  is  an  element ;  as,  Interest, 
Discount,  Equation  of  Payments,  Averaging  Accounts, 
Stocks  and  Exchange. 

Note. — lu  applying  the  Principles  of  Percentage  to  these  subjects, 
the  pupil  should  carefully  observe  what  elements  or  parts  are  given  and 
what  required  in  each  example,  and  then  apply  the  corresponding  rule  or 
formula. 


200  Percentage. 

Profit    and    Loss. 

Oral     Exercises. 

471.     1.  A  man  paid  $60  for  a  watch,  and  sold  it  at  V)% 
above  the  cost ;  how  much  did  he  gain  ? 

Analysis.— He  gained  10%  of  $60.     Now  10%  of  a  number  is  yVo.  or 
yV  ;  and  iV  of  $60  is  $6.     Therefore,  etc. 

2.  If  a  man  pays  $40  for  a  cow,  and  sells  her  at  20^  advance, 
what  will  be  his  profit  ? 

3.  A  jockey  bought  a  horse  for  $80,  and  sold  it  at  a  loss  of 
h%\  how  much  did  he  lose? 

4.  A  man  having  120  acres  of  land,  bought  25^  more  ;  how 
many  acres  did  he  buy  ? 

5.  What  part  of  a  number  is  12J;^  of  it? 

6.  What  is  \2\%  of  32  ?     Of  48  ?     Of  96  ? 

7.  What  is  6^^;  of  32  ?     Of  64  ?     Of  80  ? 

8.  What  is  33^;^  of  $15  ?     Of  $50  ?     Of  $60  ? 

9.  A  merchant  sells  flannel  at  a  profit  of  10  cts.  on  a  yard, 
and  gains  12-|-.^ ;  what  is  the  cost  ? 

Analysis. — V^\fo  —  \  ;  lience,  10  cts.  =  \  the  cost ;  and  |  are  8  times 
10  cts.,  or  80  cts.,  Ans. 

10.  A  farmer  lost  $32.40  on  a  reaping  machine,  which  w^as 
33 J-^  of  the  cost ;  what  was  the  cost  ? 

11.  A  goldsmith  sold  a  watch  at  25%  profit,  and  made  $26  ; 
what  was  the  cost  ? 

12.  A  tradesman  sold  out  his   stock   of   goods   for   $2760, 
which  was  8^^  less  than  he  paid  ;  what  did  they  cost  him  ? 

13.  A  grocer  sold  strawberries  at  15  cts.  a  liter  and  made 
20%  ;  what  did  he  pay  for  them  ? 

14.  A  fruit  dealer  sold  a  barrel  of  apples  for  $1.50,  which 
was  a  loss  of  50%  ;  what  did  he  pay  for  them  ? 

15.  A  newsboy  sells  papers  at  5  cts.  apiece,  and  makes  100%  ; 
what  does  he  pay  for  them  ? 

16.  A  man  sold  his  house  for  $7500,  which  was  33^%  more 
than  he  paid  for  it ;  required  the  cost? 


Profit  and  Loss.  201 

Written     Exercises. 

472.  Profit  and  Loss  denote  the  gain  or  loss  in  business 
transactions.     They  are  calculated  by  percentage. 

The  cost  is  the  iase ;  the  per  cent  of  gain  or  loss,  the 
rate ;  the  gain  or  loss,  the  2)ercentage ;  the  selling  price,  the 
amount  or  difference. 

473.  To  find  the  Profit  op  Loss.    (Art.  460.) 

Formula. — Profit  or  Loss  =  Cost  x  Bate. 

1.  A  house  bought  for  15860  was  sold  for  23%  above  cost ; 
what  was  the  gain  ? 

2.  A  grocer  bought  a  cask  of  oil  for  $96.50,  and  retailed  it 
at  a  profit  of  6  per  c&nt ;  how  much  did  he  make  on  his  oil  ? 

3.  A  pedlar  bought  a  lot  of  goods  for  $2150,  and  retailed 
them  at  25  per  cent  advance  ;  how  much  was  his  profit  ? 

4.  A  merchant  bought  a  cargo  of  coal  for  $450,  which  he 
sold  for  12-J-  per  cent  less  than  cost ;  what  was  his  loss  ? 

5.  What  is  the  loss  on  a  piano  that  cost  $1260,  and  sold  at 
20^  loss  ? 

6.  What  was  the  gain  on  a  form  that  cost  $3585,  and  sold  at  a 
profit  of  12^/^'  ? 

7.  What  is  the  profit  on  wool  which  cost  $2538  and  sold  at 
an  advance  of  15^? 

8.  A  dealer  bought  a  quantity  of  grain  for  $1375,  and  sold 
it  for  S%  profit ;  what  amount  did  he  receive  ?     (Art.  461.) 

9.  A  young  man  having  $2750,  lost  35^  of  it  in  speculation ; 
how  much  had  he  left  ? 

10.  Bought  a  quantity  of  produce  for  $989.33,  which  I  sold 
at  20%  loss  ;  how  much  did  I  receive  for  it  ? 

11.  A  drover  bought  a  flock  of  sheep  for  $2275,  and  sold 
them  at  25*^  advance  ;  for  how  much  did  he  sell  them  ? 

12.  A  merchant  had  a  quantity  of  groceries  on  hand,  which 
cost  him  $367.13  ;  to  close  up  his  business  he  sold  them  at  15^ 
less  than  cost ;  how  much  did  he  get  for  them  ? 

13.-  A  man  bought  a  farm  for  $875,  and  was  offered  33^ 
advance  for  his  bargain  ;  how  much  was  he  offered  ? 


202  Percentage. 

14.  A  merchant  bought  a  cargo  of  cotton  for  130000 ;  the 
price  declming,  he  sold  it  at  %Y/o  less  than  cost ;  for  how 
much  did  he  sell  it  ? 

474.  To  find  the  Rate  of  Profit  or  Loss.    (Art.  464.) 

FoEMULA. — Rate  =  Profit  or  Loss  -^  Cost. 

15.  A  dealer  bought  a  span  of  horses  for  1450,  and  sold  them 
for  IGOO  ;  what  per  cent  was  his  profit  ? 

16.  A  mowing  machine  was  sold  for  $175,  which  cost  $225  ; 
what  per  cent  was  the  loss  ? 

What  is  the  rate  per  cent  profit 

17.  On  coffee  bought  at  25  cts.  and  sold  at  30  cts.  ? 

18.  On  tea  bought  at  55  cts.  and  sold  at  67  cts.  ? 

19.  On  starch  bought  at  10  cts.  and  sold  at  13  cts.  ? 

20.  On  goods  sold  at  double  the  cost  ? 

21.  On  goods  sold  at  H  the  cost  ? 

22.  A  merchant  bought  a  quantity  of  goods  for  $155.63,  and 
sold  them  for  $148.28  ;  what  per  cent  did  he  lose  ? 

23.  A  gentleman  bought  a  house  for  $3500,  and  sold  it  for 
$150  more  than  he*  gave  ;  what  per  cent  was  his  profit  ? 

24.  A  speculator  laid  out  $7500  in  land,  and  afterwards  sold 
for  $10000  ;  what  per  cent  did  he  make  ? 

25.  A  merchant  bought  $10000  worth  of  wool,  and  sold  it 
for  $12362  ;  what  per  cent,  and  how  much  was  his  profit  ? 

475.  To  find  the  Cost.    (Art.  466.) 

Formula. — Cost  =  Gain  or  Loss  -^  Rate. 

26.  The  loss  on  a  cargo  of  lumber  was  $1260,  which  was  23;^ 
of  the  cost ;  what  was  the  cost  ? 

27.  A  speculator  gained  $3748  in  land,  which  was  22;^  of 
the  cost ;  required  the  cost  ? 

28.  An  importer  made  $3900  on  a  cargo  of  goods,  wdiich 
was  16^^  of  the  cost ;  required  the  cost  ? 

29.  If  a  grocer  pays  $3584  for  a  cargo  of  flour,  for  how  much 
must  he  sell  it  to  gain  IQ^';^  ?    (Arts.  460,  461.) 


Profit  and  Loss,  203 

30.  A  mercliant  paid  18500  for  a  case  of  silks ;  at  what  price 
must  he  sell  it  to  lose  18^  ? 

31.  A  merchant  bought  butter  for  $322.75  ;  for  how  much 
must  he  sell  it  to  gain  15^  by  his  bargain  ? 

32.  Bought  tea  for  $437.50;  for  how  much  must  I  sell  it, 
to  make  1S%  by  the  operation  ? 

33.  What  is  the  selling  price  of  hay  bought  for  $845  and 
sold  at  1Q%  gain  ? 

34.  What  is  the  selling  price  of  land  costing  I18G8.25  and 
sold  at  12|-^  loss  ? 

35.  What  is  the  selling  price  of  goods  costing  82576.40  and 
sold  at  331^  profit? 

36.  What  is  the  selling  price  of  furniture  costing  $1848.75 
and  sold  at  a  loss  of  8^%'  ? 

476.  To  Find  the  Cost  from  the  Selling  Price  and  the   Rate  per 
cent  of  Profit  or  Loss. 

37.  A  manufacturer  sold  a  carriage  for  $432,  which  was  20^ 
above  cost ;  what  was  the  cost  ? 

Analysis.— $432  is  the  cost,  plus  20%  of  itself;  hence,  the  cost  was 
$432  H-  (1  +  .20)  =  $360,  Ans.     (Art.  461.) 

38.  Another  carriage  sold  for  $432,  which  was  20^  less  than 
cost ;  Avhat  was  the  cost  ? 

Analysis. — $432  is  the  cost,  minus  20  %  of  itself  ;  hence,  the  cost  was 
$432  -^  (1  -  .20)  =  $540,  Ans.     (Art.  462.)     Hence,  the 

^  ^    ,       (  SelUnq  Price  -^  (1  -}-  Rate  of  Gain). 

Formulas. — Cost  —  {  ^  ^-,.      ^  .         )^       -n  ±    \c  r     \ 

\  Setting  Price  -f-  (1  —  Kate  of  Loss). 

39.  A  farmer  sold  land  for  $86.50  a  hektar,  and  made  V2%  ; 
what  was  the  cost  ? 

40.  A  merchant  sold  a  bill  of  goods  for  $675},  and  made  lOJ^ 
profit ;  what  did  he  pay  for  the  goods  ? 

41.  A  drover  sold  cattle  for  $1750,  which  was  a  profit  of  12^^ ; 
what  did  they  cost  him  ? 

42.  A  dealer  sold  525  hektoliters  of  gTain  for  $2750,  which 
was  a  loss  of  15^ ;  what  was  the  cost  ? 


204  Percentage. 

Commission    and    Brokerage. 

477.  A  Commission  Merchant,  Agent,  or  Factor  is  a  person 
who  buys  or  sells  goods  or  transacts  business  for  another. 

478.  Commission  is  the  Percentage  allowed  the  agent  on  the 
money  invested  or  collected. 

479.  A  Broker  is  one  who  buys  and  sells  Stocks,  Bills  of 
Exchange,  etc.,  and  his  commission  is  called  Brokerage. 

480.  A  Consignment  is  Goods  sent  to  an  agent  to  sell. 
The  Consignor  is  the  person  sending  them. 

The  Consignee  is  the  person  to  whom  they  are  sent. 

481.  The  Net  Proceeds  are  the  gross  amount  of  a  business 
transaction,  mimes  the  commission  and  other  charges. 

482.  The  computation  of  commission  and  brokerage  is  the 
same  as  Percentage  ;  the  money  employed  being  the  dase ;  the 
per  cent  for  services,  the  rate  ;  the  commission,  the  percentage. 

483.  1.  Find  4^%  com.  on  sales  for  13468.     (Art.  460.) 

2.  An  agent  sold  a  house  for  17265  ;  what  was  his  commis- 
sion at  1^%  ? 

3.  Find  b^%  com.  on  375  bbl.  apples,  sold  at  $2.25  a  barrel. 

4.  Find  ^Y/o  com.  on  a  ton  of  wool,  at  87|^  cts.  a  pound. 

5.  A  commission  merchant  sold  goods  amounting  to  l>7468, 
at  b%  for  commission  and  guaranty.  How  much  did  he  re- 
ceive, and  how  much  did  he  pay  the  owner  ? 

6.  An  auctioneer  sold  a  farm  for  1^12482,  and  charged  3|-^ 
com.,  and  $50  for  advertising  it.  What  was  his  whole  bill,  and 
what  the  net  proceeds  ? 

7.  When  a  commission  of  1150,  at  QY/'c  is  received  for  goods 
sold,  what  is  the  amount  of  sales  ?     (Art.  466.) 

8.  When  the  commission,  at  ^V/o  is  $294  ? 

9.  When  the  commission,  at  6^'  is  $105  ? 

10.  When  the  commission,  at  lY/o  is  $270  ? 

11.  An  auctioneer  charged  $405  for  selling  a  saw-mill,  which 
was  1^%',  for  what  did  he  sell  it  and  what  did  the  owner  receive? 


Commission  and  Brokerage.  205 

12.  A  commission  merchant  charged  \\%  com.,  and  ?>\%  for 
guaranty  ;  he  received  1105.30.     What  were  the  net  proceeds  ? 

484.  To  Find  the  Amt.  of  Sales  from  the  Net  Proceeds  and  Rate. 

Formula. — Amount  of  Sales  ==  Net  Proceeds  h-  (1  —  Rate). 

13.  The  net  proceeds  of  goods  sold  were  $4845,  and  the 
agent  charged  2J%  commission  and  2^%  for  guaranty.  What 
was  the  amount  of  sales  ?  Ans.  $5100. 

14.  When  the  net  proceeds  are  1229,80  and  the  rate  3^,  what 
is  the  amount  of  sales  ? 

15.  My  agent  charged  1^%  commission  and  $62.40  expenses 
for  selling  my  house,  and  sent  me  $15250.  For  how  much  did 
the  house  sell  ? 

485.  To   find  the   sum   to   be   invested,  after   deducting  the  per 

cent  commission  from  the  amount  remitted. 

16.  If  $7098  are  remitted  to  an  agent  to  buy  cotton,  after 
deducting  4^^  com.,  how  much  will  be  left  to  be  invested  ? 

Analysis.— The  money  remitted  includes  both  the  operation. 

commission  and  the  investment.     The  money  invested  1.04  )  $7098 

is  100 %  of  itself,  and  100 %  +  4 %  =104 % .     Therefore,  ^      ~$6825 

$7098  ^  1.04  =  $6825,  the   money  to  be   invested. 
Hence,  the 

FoEMULA. — Su7?i  Invested  =  Remittance  ^  (1  +  Rate). 

17.  When  the  remittance  is  $1623.10,  and  the  commis- 
sion 2|-5^,  how  much  remains  to  be  invested  ? 

18.  When  the  remittance  is  $4454,  and  the  commission  2^^? 

19.  When  $4908  are  sent,  and  the  commission  is  ^%  ? 

20.  How  many  apples  at  $2  a  barrel  can  be  bought  for 
$6720.80,  after  deducting  b%  commission  ? 

21.  Sent  an  agent  $50000  to  buy  a  ship.  How  much  did  the 
owner  receive  after  deducting  1^%  commission  ? 

22.  How  many  buffalo  robes  at  $5  each  can  be  bought  with 
a  remittance  of  $2575,  after  deducting  3^  commission  ? 

23.  A  college  sent  an  agent  $10250  to  be  invested  in  a 
library ;  how  much  remained  after  deducting  2^%  com- 
mission ? 


206  Percentage, 

Insurance. 

486.  Insurance  is  security  against  loss. 

487.  Fire  Insurance  is  security  against  the  loss  of  property 
by  tire. 

488.  Marine  Insurance  is  security  against  the  loss  of  property 
at  sea. 

Note. — Risks  of  transportation  partly  by  land  and  partly  by  water,  are 
called  Transit  Insurance.  The  same  policy  often  covers  both  Marine  and 
Transit  Insurance. 

489.  Accident  Insurance  is  security  against  loss  by  accidents. 

490.  Health  Insurance  secures  a  certain  sum  during  sickness. 

491.  Life  Insurance  secures  a  stated  sum  to  the  heirs  and 
assigns  of  the  insured  in  case  of  death. 

492.  The  parties  who  agree  to  make  good  the  loss,  are  called 
Insurance  Companies  or  Underwriters. 

Note. — When  only  a  part  of  the  property  insured  is  destroyed,  the 
underwriters  are  required  to  make  good  only  the  estimated  loss. 

493.  The  Premium  is  the  sum  paid  for  insurance. 

494.  The  Policy  is  the  ivritten  contract  between  the  insurers 
and  the  insured. 

495.  Insurance  Companies  are  of  two  kinds  :  Stock  Com- 
panies and  Mtitual  Com2)anies. 

496.  A  Stock  Company  is  one  which  has  a  paid  up  capital, 
and  divides  tlie  profit  and  loss  among  its  stockholders. 

497.  A  Mutual  Company  is  one  in  which  the  losses  are 
shared  by  the  parties  insured. 

498.  Insurance  is  calculated  by  Percentage ;  the  sum  msured 
being  the  base ;  the  2)67'  cent  premium,  the  rate;  the  7;r^;;?/«?7Z 
itself,  the  percentage. 


Insurance.  207 


Mental     Exercises. 

499.     1.  How  much  must  be  paid  for  insuring  a  house  for 

$5000,  at  ^%  premium  ? 

Analysis. — Since  the  premium  is  i%,  the  sum  paid  must  be  ^%  oi 
$5000.     Now  1  %  of  $5000  is  $50,  and  |-%  is  i  of  $50,  or  $25,  Ans. 

2.  Find  the  annual  premium  of  insurance,  at  1\%  on  a  store 
and  goods  valued  at  $800. 

3.  I  paid  $8  for  insuring  $400 ;  what  was  the  rate  ? 

Analysis. — As  the  premium  on  $400  is  $8,  the  premium  on  $1  is  ^-^ 
of  $8  which  is  $.02,  or  2% ,  Ans.     (Art.  464.) 

4.  Paid  $15  for  insuring  $1000  ;  required  the  rate? 

5.  What  amount  of  insurance,  at  %%  can  be  obtained  for  $40  ? 

Analysis. — Since  2^  is  jf  „  or  5^  of  the  amount  insured,  the  premium 
is  5V  of  t^^s  amount ;  and  $40  is  5V  of  50  times  $40  or  $2000.  (Art.  466.) 


6.   What  amount  of  insurance,  at  4^  can  be  obtained  on  a 
vessel  for  $100  ? 

Written    Exercises. 

500.  1.   What  is  the  premium  at  2^^^  for  insuring  $16000  ? 

2.  What  is  the  premium  for  insuring  a  store  and  goods  valued 
at  $7500,  at  l^%  ? 

3.  What  is  the  premium  for  insuring  a  house  and  furniture 
valued  at  $65000,  at  i%  ? 

4.  If  $72  are  paid  for  insuring  $4800,  what  is  the  rate  ? 

5.  If  $420  are  paid  for  insurance  on  $18000,  what  is  the  rate  ? 

6.  If  $860  are  paid  for  insurance  of  $1720,  what  is  the  rate  ? 

7.  A  merchant  paid  $157.80  to  insure  his  store,  at  1^%', 
what  amount  did  he  insure  ? 

8.  Paid  $187  to  insure  half  the  value  of  a  ship  at  2f^;  what 
was  the  total  value  of  the  ship  ? 

501.  To  find  the  sum  to   be   insured   to  cover  the  value   of   the 

goods  and  premium. 

9.  Bought  goods  in  London  for  $7194.     What  sum  insured, 
at  'd^%  will  cover  the  value  of  the  goods  and  the  premium  ? 


208  Percentage, 

Analysis. — The  bill  is  100%  of  itself,  and  the  premium  is  Z\%  of  that 
sum;  therefore,  $7194  -  100% -S]  %  -  96|  %,  or  .9675  times  the  sum; 
now  $7194-f-.9675  =  $7435.658,  the  sum  required.     (Art.  468.) 

FoEMULA.— /S'zfw  insured  ==  Value  -f-  (1  —  Rate). 

10.  If  a  store  and  goods  are  worth  $16625,  what  sum  must 
be  insured,  at  ^%  to  cover  the  property  and  premium  ? 

11.  What  sum  must  be  insured,  at  2|%  on  a  consignment  of 
tea  which  cost  $352.50  to  cover  property  and  premium? 

12.  A  dealer  shipped  1000  bbls.  flour  worth  $6-|-  a  bbl. ;  for 
what  sum  must  he  take  out  a  policy,  at  2-|%'  to  cover  the  value 
of  the  flour  and  the  premium  ? 


Life    Insurance. 

502.  Life  Insurance  Policies  are  of  cliff ere7it  kinds,  and  the 
premium  varies  according  to  the  ex])ectation  of  life. 

503.  Life  Policies,  are  payable  at  the  death  of  the  party 
named  in  the  policy,  the  annual  premium  continuing  through 
life. 

504.  Term  Policies  are  payable  at  the  death  of  the  insured, 
if  he  dies  during  a  given  term  of  years,  the  annual  premium 
continuing  till  the  policy  expires. 

505.  Endowment  Policies  are  payable  to  the  insured  at  a 
given  age,  or  to  his  heirs  if  he  dies  before  that  age,  the  annual 
premium  continuing  till  the  policy  expires. 

Note. — The  expectation  of  life  is  the  average  duration  of  the  life  of 
individuals  after  any  specified  age. 

13.  What  premium  must  a  man,  at  the  age  of  27,  pay 
annually  for  a  life  policy  of  $4500,  at  4J^  ? 

14.  What  is  the  annual  premium  on  $5000,  at  h^%,  and  what 
Avill  it  amount  to  in  20  years  ? 

15.  A  man  took  an  endowment  policy  of  $25000  for  20  yrs., 
at  b%.    Which  was  the  greater,  the  sum  paid  or  the  sum  insured? 


Taxes,  209 


Taxes. 

506.  A  Tax  is  a  sum  assessed  upon  the  person,  property,  or 
income  of  citizens,  for  public  purposes. 

507.  A  Property  Tax  is  a  tax  upon  property. 

508.  A  Personal  Tax  is  a  tax  upon  the  person,  and  is  called 
ixp)oll  or  capitation  tax. 

Note. — The  term  poll  is  from  the  German  polle,  the  head ;  capitation, 
from  the  Latin  caput,  the  head. 

509.  Property  is  of  two  kinds,  personal  and  real  estate. 

510.  Personal  Property  is  that  which  is  movable  ;  as,  money, 
stocks,  etc. 

511.  Real  Estate  is  that  which  is  fixed ;  as,  houses  and 
lands. 

512.  Assessors  are  persons  appointed  to  make  a  list  of  taxable 
property  and  estimate  its  value  for  the  purpose  of  taxation. 

513.  Property  taxes  are  computed  by  Percentage. 
The  valuation  of  the  property  is  the  Base, 

The  tax  on  $1  is  the  Rate. 

The  net  sum  to  be  raised,  the  Percentage. 

514.  To  assess  a  Property  Tax,  when  the  sum  to  be  raised  and 

the  valuation  of  the  property  are  given. 

1.  A  tax  of  $12500  is  to  be  raised  in  a  town  the  property  of 
which  is  valued  at  $1500000,  and  there  are  250  polls,  each  taxed 
at  12;  what  is  the  rate  of  the  tax,  and  what  is  A's  tax  whose 
real  estate  is  valued  at  $6000,  and  personal  at  13000  ? 

Analysis.— The  sum  to  be  raised  is  $12500                     operation. 
less  $500  on  the  polls,  which  is  equal  to  |12000,              Town  tax  $12500 
and    $12000 -=- $1500000  =  $.008,   ^j^'/c,   or  8            ^q\\    '   ii           599 
mills.  

A's  property  is  $6000  +  |3000  =  |9000.     As  1500000  )  $12000 

he  pays  8  mills  on  $1,  on  $9000  he  pays  9000  Rate  .008 

X  .  008  =  $72,  and  $72  +  $2  (his  poll  tax)  =  $74. 
Ans.  The  rate  is  8  mills  or  xV/^>  ^^^^  ^^  *^^  $'^'^-     Hence,  the 


210 


Percentage, 


Rule. — /.  From  tlw  sum  to  he  raised  subtract  the  poll 
tax  and  divide  the  remainder  by  the  amount  of  taxable 
property ;  the  quotient  will  be  the  rate. 

II.  Multiply  the  valuation  of  each  man's  property  by 
the  rate,  and  the  product  plus  his  poll  tax  will  be  his 
entire  tax. 

Notes. — 1.  If  a  poll  tax  is  included,  the  sum  arising  from  the  polls 
must  be  subtracted  from  the  sum  to  be  raised,  before  it  is  divided  by  the 
value  of  taxable  property. 

2.  The  computation  of  taxes  may  be  shortened  by  finding  the  rate,  and 
giving  the  tax  on  $1  to  $10,  etc.,  as  in  the  following 

)       Tax    Ta  b  l  e. 


515.  Showing  the  tax  on  various  sums  at  the  rate  of  8  mills 


on 


Prop. 

Tax. 

Prop. 

Tax. 

Prop. 

Tax. 

Prop. 

Tax. 

$1 

$0,008 

$7 

$0,056 

140 

10.32 

$100 

$0.80 

2 

0.016 

8 

0.064 

50 

0.40 

200 

1.60 

3 

0.024 

9 

0.072 

60 

0.48 

300 

2.40 

4 

0.032 

10 

0.08 

70 

0.56 

400 

3.20 

5 

0.040 

20 

0.16 

80 

0.64 

500 

4.00 

6 

0.048 

30 

0.24 

90 

0.72 

1000 

8.00 

2.  Find  by  the  table  B's  tax  whose  property  is  valued  at 
$7256,  and  who  pays  for  3  polls  at  $1.50. 

3.  Find  O's  tax  on  property  valued  at  $9480,  who  pays  for 
3  polls  at  $1.25. 

4.  AVhat  is  D's  tax  on  a  valuation  of  $15676,  and  pays  for 

2  polls  at  $1.50  ? 

5.  A  tax  of  $250000  is  levied  on  a  County  whose  real  estate 
is  valued  at  $3000000,  and  has  500  polls  taxed  at  $2  each. 
Required  the  rate  of  tax,  a  tax  table  for  that  rate,  and  a  per- 
son's tax  whose  property  is  valued  at  $5250,  and  who  pays  for 

3  polls  at  $2  each. 


Duties  or  Oiisto7ns.  211 


Duties    or    Customs. 

516.  Duties  or  Customs  are  taxes  levied  upon  imported  goods 
for  revenue,  or  the  encouragement  of  home  industry. 

517.  An  Invoice  or  Manifest  containing  a  description  of  the 
goods  and  their  cost  in  the  country  from  which  they  are  im- 
ported, is  required  by  law  to  be  exhibited  to  the  Collector  of 
the  Port  on  the  arrival  of  the  ship. 

518.  Duties  are  either  Ad  valorem  or  Specific. 

519.  An  Ad  valorem  Duty  is  a  certain  per  cent  laid  on  the 
cost  of  goods  in  the  country  from  which  they  are  imported. 

520.  A  Specific  Duty  is  a  fixed  sum  laid  on  a  given  article 
or  quantity,  without  regard  to  its  value. 

521.  Before  calculating  specific  duties,  certain  allowances 
are  made  called  Tare,  Leakage,  and  Breakage, 

Tare  is  an  allowance  for  weight  of  box,  bag,  cask,  etc. 

Leakage  is  an  allowance  for  loss  of  liquids  in  casks. 

Breakage  is  an  allowance  for  loss  of  liquids  in  bottles. 

522.  1.  What  is  the  Specific  didy  on  75  hogsheads  of  alco- 
hol, at  Is.  per  gallon,  i.%  leakage  ? 

SOLUTION. 

The  number  of  gal.  =  63  x  75  =  4735  gal. 
The  leakage  at  4%  =  4725  x  .04  =  189  gal. 
The  net  gallons        =  4725  -  189  =  4536  gal. 
The  duty  at  Is.         =  4536s.  =  $1103.7222,  Arts. 

2.  What  is  the  specific  duty,  at  $0.75  a  meter,  on  150  pieces 
of  broadcloth,  each  containing  45  meters  ? 

3.  What  is  the  specific  duty  on  182  cases  of  shawls,  containing 
75  each,  at  $1.50  per  shawl  ? 

4.  What  is  the  duty,  at  5  cts.  a  pound,  on  400  sacks  of  coffee, 
each  containing  63  lb.,  the  tare  being  2^  ? 


212  Percentage, 

5.  What  is  the  Ad  Valorem  duty  on  2^  doz.  clocks,  invoiced 
at  $32.75,  and  5  doz.  watches  invoiced  at  $45J-,  at  25;?^  ? 

Analysis.— The  cost  of  30  clocks     =  $32.75  x  30  =    $982.50 
The  cost  of  60  watches  =  $45.25  x  60  =  $2715.00 

The  cost  of  both  =  $.B697.50 

The  ad  valorem  duty  at  25%  =  $8697.50  x  .25  =  $924,375 

6.  What  is  the  ad  valorem  duty,  at  33|^^,  on  150  chests  of 
tea,  each  weighing  60  lb.,  and  invoiced  at  48  cts.  a  pound,  the 
tare  being  5  lbs.  a  chest  ? 

7.  Find  the  ad  valorem  duty,  at  ^\%i  on  110  boxes  of  raisins, 
25  lb.  in  a  box,  invoiced  at  $0.12  a  pound,  the  tare  being  3^  lb. 
a  box  ? 

8.  At  Vl\%  what  is  the  ad  valorem  duty  on  5250  kilograms 
of  Eussia  iron,  invoiced  at  75  cts.  a  kilogram. 

Qu  EST!  ONS. 

444.  What  does  per  cent  mean  ?  447.  How  expressed  ?  452.  How 
change  a  common  fraction  to  a  per  cent  ?  454.  What  is  the  base  ?  455. 
The  rate  per  cent  ?  456.  The  percentage  ?  457.  The  amount  ?  458.  The 
difference  ? 

460.  How  find  the  percentage  when  base  and  rate  are  given  ?  461.  How 
find  amount  ?  462.  Difference  ?  464.  How  find  rate  from  base  and  per- 
centage ?    468.  How  find  base  from  the  rate  and  amount  or  difference  ? 

472.  What  are  profit  and  loss  ?  What  are  the  corresponding  parts  ? 
473.  How  find  the  profit  or  loss  ?  474.  The  rate  ?  475.  The  cost  ?  476. 
How  find  cost  from  selling  price  and  rate  ? 

478.  What  is  commission?  479.  Brokerage?  482.  How  calculated? 
Corresponding  parts  ?  485.  How  find  the  sum  to  be  invested  after  deducting 
commission  ? 

486.  What  is  insurance?  493.  The  x^reraium  ?  494.  Policy?  498. 
How  calculated  ?  501.  How  find  sum  to  be  insured  to  cover  loss  and 
premium  ? 

506.  What  are  Taxes  ?    513.  How  computed  ?    Corresponding  parts  ? 

516.  What  are  duties  or  customs?  519.  Ad  valorem?  520.  Specific? 
521.  What  deductions  are  made  in  specific  duties  ?  What  is  tare  ? 
Leakage  ?    Breakage  ? 


.•:t        — o---i'<— — - 

NT EREST 


(S-i^ 


■^^ 


523.     1.   How  much  must  I  pay  you  for  the  use  of  1100  for 
1  year  at  Q^c,  and  what  shall  I  owe  you  at  the  end  of  the  year  ? 

Analysis.— 6%  is  jf  o  ;  lieiice,  I  must  pay  you  jf  ^^  of  $100,  or  $6,  for 


Its  use. 


Again,  I  shall  owe  you  at  the  end  of  1  year  the  sum  borrowed  together 
with  $6  for  its  use,  and  $100 +  $6  =  $106,  the  amount  due. 

Note, — In  this  solution  four  elements  or  parts  are  considered,  called 
the  Interest,  the  Principal,  the  Per  cent,  and  the  Amount. 

Definitions. 

524.  Interest  is  the  money  paid  for  the  use  of  money. 

525.  The  Principal  is  the  money  for  which  interest  is 
paid. 

526.  Tlie  Rate  is  the  per  cent  of  the  principal,  paid  for  its 
use  1  year,  or  a  specified  time. 

527.  The  Amount  is  the  stmt  of  the  principal  and  interest. 

528.  Simple  Interest  is  the  interest  on  the  principal  only. 

529.  Legal  Interest  is  the  rate  established  by  law. 

530.  Usury  is  a  Jiigher  than  the  legal  rate. 

531.  Interest  differs  from  the  preceding  applications  of 
Percentage  only  by  introducing  time  as  an  element  in  connec- 
tion with  the  7'ate  per  cent. 

532.  The  Principal  is  the  Base ;  the  Per  cent  per  annum  is 
the  Rate ;  the  Interest  is  the  Percentage  ;  the  Sum  of  principal 
and  interest,  the  Amount. 


214 


Percentage. 


Ta  b  le. 

533.  Showing  the  legal  rates  of  interest  in  the  several  States, 
compiled  from  the  latest  official  sources. 


states. 

Rate  %. 

States. 

Bate  %. 

States. 

Rate  %. 

States. 

Rate  i. 

Ala 

Ark.... 
Arizona 

Cal 

Conn. . . 
Colo. . . . 
Dakota. 

Del 

Fla 

Ga 

Idaho. . . 
Ill 

8 
6 

10    Any*: 
7  'Any. 

6  1 

10  'Any. 

7  ;   12 

6  ' 

8  Any. 

7  Any. 
10      24 

6        8 

Ind.  .. 
Iowa.  . 
Kan 

Ky 

La 

Maine . . 
Md..  .. 

Mass . .  . 

1  Mich 

Minn. . . 
Miss.. . . 
Mo 

6 
6 
7 
6 
5 
6 
6 
6 
7 
7 
6 
6 

8 
10 
12 

8 

8 
Any. 

Any. 

to 

12 
10 
10 

Montana 
N.  H. . . . 
N.J 

N.  Mex.. 
N.  Y.... 
N.  C... 

Neb 

Nev 

Ohio.... 
Oregon  . 

Penn 

R.I 

10 
6 
6 
6 
6 
6 
7 

10 
6 

10 
6 
6 

Any. 

12 

8 

10 

Any, 

8 
12 

Any. 

S.C 

Tenn . .  . 

!  Texas  . . 

Utah. . . . 

Vt 

Va 

W.  Va.. 
W.  T. . . . 

Wis 

Wy 

D.C.... 

7 
6 
8 

10 
6 
6 
6 

10 
7 

12 
6 

Any, 

12 

Any. 

8 

Any. 

10 
Any. 

10 

1 

534.  In  computing  interest,  a  legal  year  is   12   calendar 

months. 

Oral     Exercises. 

535.  1.  What  is  the  interest  of  140  for  1  year  at  5^  ? 

Analysis. — At  5%,  the  interest  for  1  yr.  is  -^%q  of  the  principal,  and 
yfo  of  $40  =  $2,  Ans. 

2.  What  is  the  int.  of  150  for  1  yr.  at  5^  ?    2  yr.  ?    5  yr.  ? 

3.  Of  1100  for  1  yr.  at  %  ?    At  8^  ?    At  7%  ? 

4.  Of  $200  for  2  yr.  at  7%'  ?    At  4%  ?    At  8%  ? 

5.  Of  1500  for  2i  yr.  at  6%  ?    At  6^^?     At  10^  ? 

6.  Of  $400  for  3  yr.  at  6%  ?     At  6^  ?     At  lOf^  ? 

7.  What  part  of  a  year  is  6  months  ?   4  mo.  ?  3  mo.  ?  2  mo.  ? 
1  mo.  ?    8  mo.  ?     7  mo.  ?     9  mo.  ?     10  mo.  ?    11  mo.  ?   12  mo.  ? 

8.  What  part  of  1  year's  interest  is  tlie  interest  on  the  same, 
sum  for  6  mo.  ?    For  3  mo.?     For  4  mo.  ?     For  2  mo.  ? 

9.  At  4:%,  what  is  the  interest  of  $600  for  1  yr.  and  6  mo.  ? 

10.  Calling  a  month  30  days,  what  part  of  1  mo.  is  15  days  ?' 
Is  10  days  ?     6  days  ?     5  days  ?     3  days  ?     2  days  ?     1  day  ? 

11.  If  the  interest  on  a  sum  for  1  year  is  $48,  what  is  it  for 
1  month  ?     For  3  months?     5  months?     7  months? 

*  By  special  agreement. 


no.  6  d. 

,  at  1%  ? 

$250 

Prin. 

17.50 

Int.  1  yr. 

3.1 

Yr. 

$54.25 

Int. 

$304.25 

Amt. 

Interest  215 


PROBLEM     I. 
General     Method. 

536.  To  find  the  Interest  and  Amount, when  the  Principal,  Rate, 
and  Time  are  given. 

I.  By  the  time  expressed  decimally  in  years.     (Art.  403.) 

1.  What  is  the  interest  of  $250  for  3  yr.  1  mo.  6  d.,  at 
What  is  the  amount  ? 

EXFLAKATION. 

The  given  principal 

The  int.  of  $250,  at  1%  for  1  yr.  is  $250  x  .07 
1  mo.  6  d.  =  .1  yr.  (Art.  403) ;  hence,  the  time 
Int.  for  1  yr.  $17.50  x  3.1  gives  int.  for  3.1  yr. 
The  amount  =  prin.  $250  +  $54.25  int. 
Hence,  the 

Rule. — I.  Multiply  the  principal  by  the  given  rate,  and 
this  product  by  the  tii7ve  expressed  in  years. 

II.  vddd  the  interest  to  the  principal  for  the  amount. 

2.  Find  the  interest  of  $75.36  for  1  yr.  7  mo.  18  d.  at  5%. 
What  is  the  amount  ? 

S0LUTI0N.-I75.36  X  .05  x  1.63^  (time)= $6,154,  Int.   And  $6,154+  $75.36 
=  $81,514,  Amt. 

3.  What  is  the  int.  of  $340.20,  at  6%,  for  2  yr.  8  mo.  12  d.? 
What  is  the  amount  ? 

11.  By  Aliquot  Parts.     (Art.  280.) 

EXPLANATION. 

Taking  example  first,  the  given  Principal  is  $250  Prin. 

For  1  yr.  the  int.  at  7%  is  $250  x  .07  =         17.50  Int.  1  yr. 

For  3  yr.  the  int.  is  $17.50  x  3  =       $52.50  Int.  3  yr. 

For  1  month  the  int.  is  $17.50  -4-12  =^  1.4583^    Int.  1  mo. 

For  6  d.  (4  of  30  d.)  the  int.  is  $1.4583 V  ^5  =  .2916|    Int.  6  d. 

The  entire  int.  =  S52.50  +  1.4583^  +  .2916|       =       $54.2500       Int. 
The  prin.  $250  +  $54.25  interest  =     $304.25  Amt. 


216  Percentage. 

537.  From  the  above  illustrations  we  derive  the  following 

EuLE. — Foe  oke  year. — Multiply  the  principal  hy  the 
rate. 

For  two  or   more  years. — Multiply  the  interest  for 

1  year  hy  the  number  of  years. 

.  For  months. — Tahe  the  aliquot  part  of  1  year's  interest. 
For  days. — Tahe  the  aliquot  part  of  1  month's  interest. 
The  entire  interest  is  the  sum  of  the  partial  interests. 
For  the  Amoui^t. — Add  the  interest  to  the  principal. 

Notes. — 1.    For  1  montli  take  ^^  of  the  interest  for  1   year  ;    for 

2  montlis,  ^  ;  for  3  months,  |,  etc. 

2.  For  1  day  take  3^^  ^f  the  interest  for  1  month  ;  for  2  days,  ^^ ;  for 
6  days,  \  ;  for  10  days,  i,  etc. 

3.  In  computing  interest  30  days  are  commonly  considered  a  month. 

Solve  the  following  by  either  or  both  methods  : 

4.  What  is  the  interest  of  1684  for  1  yr.  9  mo.  10  d.  at  6^? 

5.  At  ^%,  what  is  the  amt.  of  11125  for  1  yr.  2  mo.  3d.? 

6.  At  5^,  what  is  the  amt.  of  $1056  for  10  mo.  24  d.  ? 

7.  At  6^,  what  is  the  int.  of  $1340  for  1  mo.  15  d.  ? 

8.  At  7^,  what  is  the  int.  of  $815  for  3  yr.  2  mo.  21  d.  ? 

9.  At  m,  what  is  the  amt.  of  $961  for  2  yr.  4  mo.  10  d.  ? 

10.  AVhat  is  the  int.  of  $3500  for  11  mo.  20  d.,  at  10^? 

11.  What  is  the  amt.  of  $39,275  for  2  yr.  6  mo.,  at  12^^? 

12.  What  is  the  int.  of  $113.61  for  5  yr.  5  mo.,  at  5%'  ? 

13.  What  is  the  int.  of  $1000  for  2  yr!^  3  mo.  10  d.,  at  4J^? 

14.  What  is  the  int.  of  $1260.34  for  10  yr.,  at  ?>%  ? 

15.  What  is  the  int.  of  $234.56  for  2  yr.  4  mo.  5  d.,  at  6;^  ? 

16.  What  is  the  amt.  of  $600  for  1  yr.  6  mo.  10  d.,  at  h%  ? 

17.  Find  the  amount  of  $60  for  7  mo.,  at  %%. 

18.  What  is  the  interest  of  $96  for  10  months,  at  ^%  ? 

19.  At  ^%,  what  IS  the  amt.  of  $700  for  1  yr.  2  mo.  12  d.? 

20.  At  4^,  what  is  the  amt.  of  $470  for  10  days? 

21.  Find  the  int.  of  $1000  for  1  yr.  1  mo.  1  d.,  at  %%. 

22.  At  6^,  what  is  the  amt.  of  $4565.61  for  4  mo.  7  days  ? 


Interest.  217 

23.  What  is  the  interest  of  15625.43  for  4  mo.  18  d.,  at  6J^  ? 

24.  At  54^,  what  is  the  int.  of  $624,625  for  7  mo.  3  days? 

25.  At  S%,  what  is  the  int.  of  $11261.18f  for  3  mo.  3  days? 

26.  At  7^0  what  is  the  amt.  of  89208.95  for  11  mo.  5  days  ? 

27.  What  is  the  amt.   of  $15206.843,  at  1\%,  for   1   year 
8  months  25  days? 

28.  The  amtf  of  $10050.69,  at  ^%,  for  2  yr.  9  mo.  5  d.  ? 

29.  What  is  the   amt.  of   811607.858,  at   7^,   for   3  years 
6  months  9  days  ? 

30.  The  amt.  of  $41361.18,  at  6^,  for  5  yr.  7  mo.  3  d.  ? 

31.  What  is  the  interest  on  $1145  from  July  20th,  1881,  to 
Dec.  7th,  1881,  at  7^^?     (Art.  409,  K  2.) 

Note.— The  time  is  4  mo.  and  11  d.  (July)  +  7  d.  (Dec.)  =  4  mo.  18  d. 

32.  What  is  the  interest  on  a  note  of  $568.45  from  May  21st, 
1881,  to  March  25th,  1882,  at  b%  ? 

33.  Required  the  amount  of  $2576.81  from  Jan.  21st,  1881, 
to  Dec.  18th,  1881,  at  1%. 


Six     Per     Cent    Method. 
Develofment    of    Pbincip l es, 

538.  The  interest  of  $1  at  6% 

For  1  yr.,     or  12  mo.,  is  6  cts.,  =  .06  of  the  principal. 

For  -J-  yr.,     or  2  mo.,  is  1  cent,  ■—  .01  of  the  principal. 

For  ^  yr.,  or  1  mo.,  is  5  m.,  =  .005  of  the  principal. 

For  \  mo.,   or  6  d.,  is  1  m.,  =  .001  of  the  principal. 

For  ^Q  mo.,  or  1  d.,  is  -J  m.,  =  .000|-  of  the  principal. 

Hence,  we  derive  the  following 

Principles. 

i°.   Tlie  interest  of  SI  at  6%,  is  half  as  many  cents  as  there 
are  montJis  m  the  given  time. 

2°,   The  interest  of  $1  at  6%,  is  one-sixth  as  7nany  mills  as 
there  are  days  in  the  given  time. 

10  « 


218  Percentage. 

539.  To  find  the  Interest,  when  the  Principal,  Rate,  and  Time 

are  given. 

1.  What  is  the  interest  of  $250.26  for  1  yr.  3  mo.  21  d., 

at  6^  ?     What  is  the  amount  ? 

ExPLA. — The  interest  of  $1  for  15  mo.  =  .075  operation. 

By  2°,  int.  of  $1  for  21  d.       =  .0035  1250.26   Prin. 

Int.  of  $1  for  1  yr.  3  mo.  21  d.  =  .0785  .0785  Int.  $1. 

As  tlie  interest  of  $1  for  the  given  time  and  125130 

rate  is  $.0785,  the  interest  of  $250.26  must  be  o  00208 
$250.26  X  .0785  =  $19.04541  interest. 

The  prin.  $250.26  +  $19.64541  =  $269.90541, 


17.5182 


Amount.     Hence,  the  $19.645410,    Ans. 

Rule. — Multiply  the  principal  hy  the  interest  of  $1 
for  the  ^iven  time,  and  rate. 

Notes. — 1.  When  the  rate  is  greater  or  less  than  6%,  find  the  interest 
of  the  principal  at  6%  for  the  given  time  ;  then  add  to  or  subtract  from  it 
such  a  part  of  itself,  as  the  given  rate  exceeds  or  falls  short  of  6  per  cent. 

2.  If  the  mills  are  5  or  more,  it  is  customary  to  add  1  to  the  cents  ;  if 
less  than  5,  they  are  disregarded. 

3.  Only  three  decimals  are  retained  in  the  following  Answers,  and  each 
answer  is  found  by  the  rule  under  which  the  Example  is  placed. 

4.  In  finding  the  interest  of  $1  for  days,  it  is  sufficient  for  ordinary 
purposes  to  carry  the  decimals  to  four  places. 

2.  What  is  the  amt.  of  1350.60  for  1  yr.  5  mo.  15  d,,  at  6%  ? 

3.  What  is  the  int.  of  $56.19  for  4  mo.  3  d.,  at  '7%  ? 

4.  What  is  the  int.  of  $242.83  for  7  mos.  18  d.,  at  o%  ? 

5.  Find  the  int.  of  $781.13  for  11  mo.  21  d.,  at  6^. 

6.  Find  the  int  of  $968.84  for  2  yr.  10  mo.  26  d.,  at  6%. 

7.  What  is  the  int.  of  $639  for  18  mo.  29  d.,  at  1%? 

8.  Wliat  is  the  int.  of  $745.13  for  17  d.,  at  5%  ? 

9.  What  is  the  int.  of  $1237.63  for  8  mo.  3  d.,  at  S%  ? 

10.  What  is  the  int.  of  $2046^  for  25  d.,  at  4:%  ? 

11.  Find  the  amount  of  $640.37^  for  9  mo.  15  d.,  at  10^. 

12.  Find  the  amount  of  $2835.20  for  2  mo.  3  d.,  at  9^^ 

13.  Find  the  amount  of  $4356.81  for  3  mo.  10  d.,  at  6^%. 

14.  What  is  the  int.  of  $12240  for  63  d.,  at  ^%  ? 

15.  What  is  the  int.  of  $350000  for  10  d.,  at  di%  ? 


Interest.  219 

Method     by     Days. 

540.  1.   What  is  the  interest  of  I248.G0  for  93  days,  at  Q%  ? 

OPERATION. 

Analysis.— Since  the  interest  for  30  1248.60     Priu. 

days  is  5  mills,  or  -^^^  of  the  principal,  93      No.  d. 

for  1  day  it  is  3V  of  ^^^,  or  « oV 0  '  lience,  74580 

tlie  interest  for  93  days  is   gfgo  of  tlie  99q'*'4.0 

principal.    And  ^ff^  of  $248. 60 =($248.60  j^^rf^U_ 

X  93)-^6000  =  $3.85.     Hence,  the  6|000  )  23|119.80 

$3,853,  Ans, 

EuLE. — Multiply  the  principal  hy  the  number  of  clays, 
and  divide  the  product  hy  6000. 

2.  Find  the  interest  of  $360  for  95  d.,  at  1%.     Ans.  $6.65. 

What  is  the  interest  of 

3.  $450  for  63  d.  at  6^  ?  7.  $600  for  63  days  at  b%  ? 

4.  $245.50  for  33  d.  at  Q%  ?  8.  $735  for  45  days  at  7%? 

5.  $278.68  for  75  days  at  Q%  ?  9.  $1200  for  60  d.  at  b%  ? 

6.  $500.75  for  130  days  at  Q%  ?  10.  $1500  for  93  d.  at  8%? 

Exact     Inter  est. 

541.  The  methods  based  upon  the  supposition  that  360  days 
make  a  year  and  30  days  a  month,  though  common,  are  not 
strictly  accurate.  As  a  year  contains  365  days,  the  int.  found 
by  these  methods  is  ^-|-g^,  or  -^  part  of  itself  too  large.     Hence, 

542.  To  compute  exact  interest  for  months  and  days, 
find,  the  interest  hy  the  6%  method  and  subtract  from 
it  yV  part  of  itself.    (An.  Int.,  Art.  905,  App.) 

1.  What  is  the  exact  interest,  at  6^,  of  $248.60  for  3  mo.  3  d.  ? 
Ans.  The  interest  at  Q%  is  $3,853,  J^  part  of  which  is  $.053, 

and  $3.853— $.053  =  $3.80. 

2.  What  is  the  exact  interest  of  $2568  for  93  d.,  at  6^? 

3.  What  is  the  exact  interest  0^  $5000  for  12  d.,  at  1%  ? 


220  Percentage, 


Partial    Payments. 

543.  Partial  Payments  are  parts  of  a  note  paid  at  different 
times. 

544.  A  Promissory  Note  is  a  written  promise  to  pay  a  speci- 
fied sum  at  a  given  time. 

545.  The  Maker  is  the  person  who  signs  the  note. 

546.  The  Payee  is  the  person  to  whom  it  is  to  be  paid. 

547.  The  Holder  is  the  person  who  has  the  note  in  liis 
possession. 

548.  Indorsements  are  partial  payments,  the  amount  and 
date  of  which  are  written  upon  the  back  of  notes  and  bonds. 

549.  The  Face  of  a  note  is  the  sum  named  in  it. 

550.  A  Negotiable  Note  is  one  payable  to  the  bearer,  or  to 
the  order  of  the  person  named  in  it. 

Notes. — 1.  A  note  payable  to  A.  B.,  or  "order,"  is  transferable  by 
indorsement ;  if  to  A.  B.,  or  "  bearer,"  it  is  transferable  by  delivery. 
Treasury  notes  and  bank  bills  belong  to  this  class. 

2.  If  the  words  "  order  "  and  •'  bearer "  are  both  omitted,  the  note  can 
be  collected  only  by  the  party  named  in  it. 

551.  An  Indorser  is  a  person  who  writes  his  name  on  the 
back  of  a  note  as  security  for  its  payment. 

552.  The  Maturity  of  a  note  is  the  day  it  becomes  legally 
due.  In  most  States  a  note  does  not  mature  until  3  days  after 
the  time  named  for  its  payment. 

These  three  days  are  called  Days  of  Grace. 

553.  To  compute   Interest  on   notes  and   bonds,  when  JPartial 

l\ujnient.s  have  been  made. 

United     States     Rule. 

Find  the  amount  of  the  piineipal  to  the  tUne  of  the 
first  payment,  and  suhtracting  the  payment  from  it,  find 
the  amount  of  the  remainder  as  a  new  principal,  to  the 
time  of  the  next  paymenl. 


Partial  Payments.  221 

//  the  payment  is  less  than  the  interest,  find  tlie 
amount  of  the  principal  to  the  time  when  the  sum  of 
the  payments  equals  or  exceeds  the  interest  due;  and 
subtract  the  sum  of  the  payments  from  this  amount. 

Proceed  in  this  manner  to  the  tUi%e  of  settlement. 

Notes. — 1.  The  principles  upon  which  the  preceding  rule  is  founded 
are,  1st.  That  payments  must  be  applied  first  to  discharge  accrued 
interest,  and  then  the  remainder,  if  any,  toward  the  discharge  of  the 
principal. 

2d.  That  only  unpaid  principal  can  draw  interest. 

3.  The  following  examples  show  the  common  forms  of  promissory 
notes.  The  first  is  negotiable  by  indorsement ;  the  second  by  delivery ;  the 
third  is  o,  joint  note,  but  not  negotiable. 

^'^50.  Washington,  Jan.  1st,  1880. 

1.  On  demand  I  promise  to  pay  to  the  order  of  Alexander 
Hunter,  eight  hundred  fifty  dollars,  loith  interest  at  6  per 
cent,  value  received.  John  FRAXKLiiT. 

The  following  payments  were  endorsed  on  this  note : 

July  1st,  1880,  received  $100.62. 
Dec.  1st,  1880,  received  $15.28. 
Aug.  13th,  1881,  received  1175.75. 

What  was  due  on  taking  up  the  note  Jan  1st,  1882  ? 

SOLUTION. 

Principal,  dated  Jan.  1st,  1880,  $850.00 

Int.  to  1st  payt.  July  1st,  1880  (6  mo.)  (Art.  539),  _^^^ 

Amoimt,  =    875.50 

1st  payment,  July  1st,  1880,  100.62 

Remainder,  or  new  principal,  =     774.88 

Int.  from  1st  payt.  to  Dec.  1st  (5  mo.)  19.37 

2d  payt.  less  than  int.  due,  $15.28 

Int.  on  same  priu.  to  3d  payt.,  Aug.  18  (8  mo.  12  d.)  32,54 

Amou7it,  =     826.79 

3d  payt.,  to  be  added  to  2d,  |175.75     =     19103 

Remainder,  or  new  principal,  =     635.76 

Int.  to  Jan.  1st,  1882  (4  mo.  18  d.)  14.62 

Balance  due  Jan.  1st,  1882,  =  $650.38 


222  Percentage, 

$692j%%.  Boston,  Aug.  15th,  1879. 

2.  Three  months  after  date,  I  pro7nise  to  imy  John  War- 
ner, or  hearer,  six  hundred  and  ninety-two  dollars  and  thirty- 
five  cents,  with  interest  at  6  yer  cent,  value  received. 

Samuel  Johj^son". 

Endorsed  Nov.  loth,  1879,  $250,375. 
Endorsed  March  1st,  1880,  $65,625. 

How  much  was  due  on  the  note,  July  4th,  1881  ? 


?^_:  New  York,  May  10th,  1878. 

3.  For  value  received,  we  jointly  and  severally  'promise  to  pay 
James  Monroe  &  Sons,  five  hundred  dollars  on  demand, 
with  interest  at  7  per  ce?it. 

George  Johkson. 

Henry  Smith. 

The  following  sums  were  endorsed  upon  it : 

Received,  Nov.  10th,  1878,  175. 
Received,  March  22d,  1879,  $100. 

"What  was  due  on  taking  up  the  note,  Sept.  28th,  1879  ? 


^^0^^'  Philadelphia,  June  20th,  1878. 

4.  Six  months  after  date,  I  promise  to  pay  3fessrs.  Caret, 
Hart  S  Co.,  or  order,  one  thousand  dollars,  luith  interest  at 
5  per  cent,  value  received. 

Horace  Preston. 

Endorsed  Jan.  10th,  1879,  $125. 
Endorsed  June  16th,  1879,  $93. 
Endorsed  Feb.  20th,  1880,  $200. 

"What  was  the  balance  due  on  the  note,  Aug.  1st,  1880  ? 

Note. —  Massachusetts,  New  York,  Pennsylvania,  Ohio,  Illinois,  and 
most  of  the  other  States  have  adopte  1  this  rule.  (For  Connecticut,  Ver- 
mont, and  New  Hampshire  methods,  see  Art  906-908,  Appendix.) 


Fartial  Payments,  223 


Mercantile     Method. 

554.  When  Partial  Payments  are  made  on  sliort  notes  or 
interest  accounts,  business  men  commonly  employ  the  follow- 
ing method  : 

Find  the  amount  of  the  whole  debt  to  the  time  of  set- 
tlement;  cdso  find  the  amount  of  each  payment  fi^om 
the  time  it  was  made  to  the  time  of  settlement. 

Subtract  the  amount  of  the  payments  from  the  amount 
of  the  debt ;  the  remainder  will  be  the  balance  due. 

^i^  Albany,  March  31st,  1880. 

5.  On  demand,  I  'promise  to  pay  to  the  order  of  Henry 
Pattox,  four  hundred  and  sixteen  dollars,  with  interest  at 
7  ])er  cent,  value  received. 

JoHiT  Marshall. 

Eeceived  on  the  aboye  note  the  following  sums : 

June  15th,  1880,  $35.00. 

Oct.  9th,  1880,  123.00. 

Jan.  12th,  1881,  $68.00. 
What  was  due  on  the  note,  Sept.  21st,  1881  ? 

SOLUTION. 

Principal,  dated  March  21st,  1880,  $416,000 

Int.  to  settlement  (1  yr.  6  mo.),  o.tl'/o,  43.680 

Amount,  Sept.  21st,  1881,  =     459.680 

1st  payt,  $35.00,  Time  (1  yr.  3  mo.  6  d.),  Amount  =  $38,103 

2d  payt.,  $23.00,  Time  (11  mo.  12  d.),  Amount  =  24.530 

3d  payt.,  $68.00,  Time  (8  mo.  9  d.).  Amount  =  71.292 

Amount  of  the  payments,  =     133.925 

Balance  due  Sept.  21st,  1881,  .  $325,755 

6.  A  bill  of  goods  amounting  to  $750,  was  to  be  paid  Jan. 
1st,  1880.  Received  June  10th,  $145  ;  Sept.  23d,  $465  ;  Oct. 
3d,  $23 ;  what  was  due  on  the  bill  Dec.  31st,  1880,  int.  Q%  ? 

7.  An  account  of  $1200  due  March  3d,  received  the  follow- 
ing payments:  June  1st,  $310 ;  Aug.  7th,  $219;  Oct.  17th, 
$200  ;  what  was  due  on  the  27th  of  the  following  Dec,  allowing 
7^  interest  ? 


224  Percentage, 


PROBLEM     II. 

555.  To  find  the  Hate,  when  the  Principal,   Interest,  and   Time 

are  given. 

1.  At  what  rate  of  interest  must  1236  be  loaned,  to  gain 
$17.70  in  1  year  and  3  months? 

Analysis.— The  int.  of  $236  for  1  yr.  at  1  %  =  $236  x  .01         =     $3.36 
The  int.  for  3  mo.  (|  yr.)  =  $2.36  x  i  =         .59 

The  int.  for  1  yr.  3  mo.  at  Ifc  =     $2.95 

Now  as  $2.95  gain  requires  1%,  $17.70  gain  requires  as  many  per  cent 
as  $2.95  are  contained  times  in  $17.70,  or  6%,  Ans.     Hence,  the 

EuLE. — Divide  the  given  interest  hy  the  interest  of  the 
principal,  at  1  per  cent  for  the  time. 

Formula. — Rate  =  Interest  -f-  {Prin.  x  1%  x  Time). 

Note. — When  the  amount  is  given  the  principal  and  interest  may  be 
said  to  be  given.  For,  the  amt.  =  the  prin.  +  int. ;  hence,  amt.  —int.  =  the 
prin. ;  and  amt.  —prin.  =  the  int. 

2.  At  what  rate  per  cent,  must  1450  be  loaned,  to  gain  $56.50 
interest  in  1  year  and  6  months  ? 

3.  At  what  per  cent  must  $750  be  loaned,  to  gain  1225  in 
4  years  ? 

4.  A  man  has  $8000  which  he  wishes  to  loan  for  $500  per 
annum ;  at  what  per  cent  must  he  loan  it  ? 

5.  A  gentleman  deposited  $1250  in  a  savings  bank,  for  which 
he  received  $31.25  every  6  months;  what  per  cent  interest  did 
he  receive  on  his  money  ? 

6.  A  capitalist  invested  $9260  in  railroad  stock,  and  drew  a 
semi-annual  dividend  of  $416.70  ;  what  rate  per  cent  interest 
did  he  receive  on  his  money  ? 

7.  A  man  built  a  hotel  costing  1175000,  and  rented  it  for- 
18750  per  year  ;  what  per  cent  int.  did  his  money  yield  him  ? 

8.  A  man  gave  his  note  payable  in  1  year  and  3  montlis  for 
$640,  and  at  its  maturity  paid  1688  ;  what  was  the  rate  of 
interest  ? 

9.  At  what  rate  must  1865  be  loaned  for  2  years  to  yield 
$129.75  interest? 


Interest,  325 


PROBLEM     III. 

556.  To  find  the    Time  when  the   Principal,  Interest,  and  Rate 

are  given. 

1.  In  what  time  will  $500  gain  145  at  6%  ? 

Analysis.— The  interest  of  $500  for  1  yr.  at  6%  is  $30.  opbbation. 

Therefore,  to  gain  $45  will  require  the  same  principal  as  30  )  $45.00 

many  years  as  $30  are  contained  times  in  $45  ;  and  $45  h-  <  ~    7"^" 
$30  =:  1.5  or  1|  years,  Ans.     Hence,  the 

EuLE. — Divide  the  given  interest  by  the  interest  of  the 
principal  for  1  year,  at  the  given  rate. 

Formula. — Time  =  hit.  -^  {Prin.  x  Rate). 

Notes. — 1.  If  the  quotient  contains  decimdls,  reduce  them  to  months 
and  days.     (Art.  402.) 

2.  If  the  amount  is  given  instead  of  the  principal  or  the  interest,  iSnd 
the  part  omitted,  and  proceed  as  above. 

3.  At  100,^^,  any  sum  will  double  itself  in  1  year;  therefore,  any  per 
cent  will  require  as  many  years  to  double  the  principal,  as  the  given  per 
cent  is  contained  times  in  100%! 

2.  In  what  time  will  $4500  gain  $430  at  b%  ? 
■    3.  How  long  will  it  take  $5000  to  earn  $5000  at  Q>%  ? 

4.  How  long  will  it  take  any  sum  to  double  itself  at  4^? 

5^?     6^?     7^?     10^? 

PROBLEM     IV. 

557.  To  find  the  Principal,  when  the  Interest,  Rate,  and  Time 

are  given. 

1.  What  principal  at  Q%  will  yield  $225  interest  in  2  yr.  6  mo.  ? 

Analysis.— At  6  % ,  the  interest  of  |1  for  2  yr.  6  mo.  operation. 

is  $.15,  therefore,  $225   must  be  the  int.  of  as  many  .15  )  225.00 

dollars  as   $.15   are   contained    times  in    $225,   and  .         ^i^nr 

$225 --$.15  =  $1500,  ^;?«.     Hence,  the  Ans.   ^l0^(} 

Rule. — Divide  the  given  interest  hij  the  interest  of  $1 
for  the  given  time  and  rate,  expressed  deeiniaUy. 

Formula. — Principal  =  Interest  h-  {Pate  x  Time). 


226  Percentage. 

2.  What  principal  at  '7%  will  yield  1500  in  1  year  ? 

3.  At  6%  what  principal  will  yield  1350  in  6  months  ? 

4.  What  principal  at  5%  will  yield  $400  in  7  mo.  15  d.  ? 

5.  What  sum  must  a  father  invest  at  6%,  that  his  son,  now 
18  yr.  old  may  have  1^5000  when  he  is  21  ?     (Art.  556,  N.  2.) 

6.  What  sum  loaned  at  1%  a  mo.  will  amount  to  $500  in  1  yr.? 

7.  AYhat  sum  must  be  loaned  at  4:%  a  year  to  amount  to 
11200  in  8  months  ? 


Compound    Interest. 

558.  Compound  Interest  is  the  interest  of  the  jyrmcipal  and 
of  the  unpaid  interest  after  it  becomes  due. 

559.  To  compute  Compound  Interest,  when  the  Principal,  Rate 

and  time  of  compounding  it  are  given. 

I.  What  is  the  compound  interest  of  1500  for  3  years  at  6%? 

Principal,  '  =      $500 

Int.  for  1st  year,  $500  x  .06,  30 

Amt.  for  1  yr.,  or  2d  prin.,  =        530 

Int.  for  2d  year,  $530  x  .06,  31.80 

Amt.  for  2  yr.,  or  3d  prin.,  =         561.80 

Int.  for  3d  year,  $561.80  x  .06,  33.71 

Amt.  for  3  years,  =        595.51 

Original  principal  to  be  subtracted,  500 

Compound  int.  for  3  years,  =  95.51 

Hence,  the 

EULE. — I.  Find  the  amount  of  the  principal  for  the 
first  period.  Treat  this  amount  as  a  new  principal,  and 
find  the  amount  due  on  it  for  the  next  period,  and  so  on 
through  the  juhole  time 

II.  Subtract  the  given  principal  from  tlie  last  amount, 
and  the  rem^ainder  will  he  the  com  pound  interest. 

Note. — If  there  are  months  or  days  after  the  last  regular  period  at 
which  the  interest  is  compounded,  find  the  interest  on  the  amount  last 
obtained  for  them,  and  add  it  to  the  same,  before  subtracting  the  principal. 


Compound  Interest. 


227 


2.  What  is  the  compound  int.  of  S450  for  3  yr.  6  mo.,  at  6^  ? 

3.  What  is  the  compound  int.  of  1550  for  3  yr.  4  mo.,  at  1%  ? 

4.  What  is  the  compound  int.  of  1850  for  4  yr.  6  mo.,  at  b%  ? 

5.  What  is  the  com.  int.  of  1865  for  5  yr.,  at  1%  ? 

6.  What  is  the  amt.  of  1950  for  6  yr.  3  mo.,  at  b%,  com.  int.? 

560.  Table  shoiving  the  amount  of  II,  at  3,  3  J,  4,  5,  and  Q% 
compound  interest,  for  any  number  of  years  from  1  to  20. 


Yrs 

3%. 

3i%- 

4%. 

5%. 

6%. 

I 

1.030  000   I 

035  000 

1.040  000 

1.050  000 

1.060  000 

^ 

1.060  900   I 

071  225 

1. 08 1  600 

1.102  500 

1. 123  600 

3 

1.092  727   I 

108  718 

I. 124  864 

1.157  625 

I. 191  016 

4 

I. 125  509   I 

147  523 

1-169859 

1. 215  506 

1.262  477 

5 

1. 159  274   I 

187  686 

1. 216  653 

1.276  282 

1.338  226 

6 

1. 194052   I 

.229255 

1.265  319 

1.340096 

1.418519 

7 

1.229  ^74   I 

.272  279 

^•315  932 

1.407  100 

1-503630 

8 

1.266  770   I 

316  809 

1.368569 

1-477  455 

1.593848 

9 

1.304773   I 

362  897 

1.423  312 

1-551 Z^^ 

1.689479 

lO 

1.343  916   I 

410599 

1.480  244 

1.628895 

1.790  848 

II 

1.384234   I 

459970 

1-539  451 

1. 710  339 

1.898  299 

12 

1.425  761   I 

511  069 

1. 601  032 

1-795  856 

2.012  196 

13 

1.468534  I 

563956 

1.665  074 

1.885649 

2.132  928 

14 

1. 512  590   I 

618695 

1. 731  676 

1.979932 

2. 260  904 

^5 

1-557967   r 

675  349 

1.800  944 

2.078  928 

2-396558 

16 

1.604  706  I 

733  986 

1.872  981 

2.182  875 

2-540352 

17 

1.652848  I 

794676 

1.947  900 

2. 292  018 

2.692  773  ' 

18 

1.702433   I. 

857489 

2.025  8^7 

2.406  619 

2.854339  , 

19 

1-753506  ;  I 

922  501 

2.106  849 

2.526  950 

3.025  600 

20 

1.806  in  1  I. 

i 

989  789 

2. 191  123 

2.653  298 

3.207  135 

Note. — Compound  interest  cannot  be  collected  by  Imo  ;  but  a  creditor 
may  receive  it,  without  incurring  the  penalty  of  usury.  Savings  Banks 
pay  it  to  all  depositors  who  do  not  draw  their  interest  when  due. 


228  Percentage, 

561.     1.  What  is  the  int.  and' ami  of  12000  for  10  yr.  at  ^6%  ? 

Solution.— Tabular  amt.  of  $1  for  10  yr.  at  3%,  $1.34^916x2000 
=  $2687.832,  amt.  for  10  yr.  And  $2687. 832 -$2000  prin.  =  $687,832, 
Com.  Int.  for  10  years.     Hence,  the 

EuLE. — ^I.  Multiply  the  tabular  amount  of  $1  for  the 
given  time  and  rate  by  the  pidncipal ;  the  product  will 
he  the  amount. 

II.  From  the  amount  subtract  the  principal,  and  the 
remainder  will  be  the  compound  interest. 

Notes. — 1.  If  the  given  number  of  years  exceed  that  in  the  Table,  find 
the  amount  for  any  convenient  period,  as  half  the  given  years;  then  on  this 
amount  for  the  remaining  period. 

3.  If  interest  is  compounded  semi-annually  take  |  the  given  rate  and 
twice  the  number  of  years  ;  if  compounded  quarterly,  take  ^  the  given 
rate  and  4  times  the  number  of  years. 

2.  What  is  the  amt.  of  13500  for  G  yr.,  at  5^  com.  interest  ? 

3.  What  is  the  amount  of  $350  for  12  years,  at  4^  ? 

4.  What  is  the  com.  int.  of  $469  for  15  years,  at  3%  ? 

5.  What  is  the  com.  int.  of  $500  for  24  years,  at  6%  ? 

6.  What  is  the  com.  int.  of  $650  for  30  years,  at  3^%? 

7.  What  is  the  amount  of  $1000  for  3  yr.,  at  (]%  compound 
interest,  payable  semi-annually  ? 

8.  What  is  the  amount  of  $1200  for  2  years,  at  12^  componnd 
interest,  payable  quarterly  ? 

9.  What  is  the  amt.  of  $1500  for  5  yr.  3  mo.,  at  5%  com.  int.  ? 


Questions. 

524.  What  is  interest  ?  525.  Principal  ?  526.  Rate  ?  527.  Amount  V 
528.  What  is  simple  int.  ?     529.  Legal  interest  ?     530.  Usury? 

536.  The  general  method  of  computing  interest  ?  539.  The  Q^i  method  ? 
540.  The  method  by  days  ?     542.  Exact  interest  ? 

543.  What  are  Partial  payments?  544.  Promissory  note?  549.  The 
face  of  a  note  ?  550.  A  negotiable  note  ?  552.  The  maturity  of  a  note  ? 
548.  Indorsements?     553.  U.  S.  Rule  for  partial  payments? 

555.  How  find  the  rate  ?  556.  The  time  ?  557.  The  principal  ?  558. 
Compound  Interest  ?    559.  How  computed  ? 


ISCOUNT. 


-^^- 


Oral     Exercises. 

562.     1.  The  price  of  a  watch  was  150,  but  for  cash  it  was 
sold  at  10^  off  ;  how  much  was  the  deduction  ? 

Analysis. — 1%  of  $50  is  50  cents,  and   10;^   is  10  times  .50,  or  $5; 
hence  the  deduction  was  $5,  Ans. 

2.  A  man  asked  1200  fpr  a  horse,  but  for  cash  would  take 
6%  off  ;  liow^  much  was  deducted  ? 

3.  A  merchant  sold  a  bill  of  goods  amounting  to  $500,  and 
for  cash  deducted  6^}' ;  how  much  was  deducted  ? 

4.  A  man  owed  $800  on  Acct.,  and  settled  it  for  cash  at  4^ 
off  ;  what  was  the  deduction  ? 

5.  If  you  borrow  1300  and  pay  G^  in  advance  for  its  use, 
how  much  is  deducted  from  the  loan  ? 


True    Discount. 

563.  Discount  is  a  deduction  from  a  stated  price,  or  from  a 
debt  paid  before  it  is  due. 

564.  True  Discount  is  the  difference  between  the  face  of  a 
debt  and  its  present  wortli. 

The  Present  Worth  of  a  debt,  due  at  some  future  time  with- 
out interest,  is  the  sum  which  put  at  interest  at  the  legal 
rate  will  amount  to  the  debt  when  it  becomes  due. 

565.  To  find  the  Present  Worth  and  True  Discount. 

1.  What  is  the  present  worth  and  true  discount  of  $378,  due 
in  1  year  and  8  months,  at  G;^  ? 


230  Percentage, 

Analysis. — The  amount  of  $1,  at  6%,  for  1  yr.  8  mo.  =  $1.10.  Since 
$1.10  is  the  amt.  of  $1,  at  6^  for  the  given  time,  $378  is  the  amt.  of  as 
many  dollars  for  the  same  time  and  rate,  as  $1.10  is  contained  times  in 
$378.  and  $378^$1.10  =  $343.64,  present  worth.  Then  $378-  $343.64 
=  $34.36,  the  true  discount.     Hence,  the 

EuLE. — I.  Divide  the  debt  by  the  cunount  of  $1  for  the 
given  time  and  rate;  the  quotient  will  be  the  present  worth, 

II.  Subtract  the  present  worth  from  the  debt,  and  the 
reinainder  will  be  the  true  discount. 

Find  the  present  worth  and  true  discount  of 

2.  $850.25,  due  in  \\  years,  at  6^. 

3.  $1272.50,  due  in  1  yr.  3  mo.,  at  7^. 

4.  $2895,  payable  in  2  years,  at  h%, 

5.  $5650.75,  payable  in  3|-  years,  at  ^\%, 

6.  $10000,  due  in  1  yr.  5  mo.,  at  '^%. 

7.  What  is  the  difference  between  the  interest  and  true  dis- 
count of  $12250,  for  1  year,  at  ?)%  ? 

8.  Bought  a  farm  for  $4822,  payable  in  2-|-  years  without 
interest,  but  for  cash  20^  discount ;  what  was  the  true 
discount  ? 

9.  When  money  .is  worth  5*^,  which  is  preferable,  $12000 
cash,  or  $13000  payable  in  1  year  ? 


Bank    Discount. 

566.  Bank  Discount  is  simple  interest,  paid  in  advance. 

567.  The  Proceeds  of  a  note  are  the  part  paid  to  the  owner  ; 
the  Discount  is  the  part  deducted. 

568.  The  Maturity  of  a  note  is  on  its  last  day  of  grace. 

Note. — If  the  last  day  of  grace  occurs  on  Sunday  or  a  legal  holiday,  the 
note  matures  on  the  preceding  day. 

569.  The   Term   of   Discount   is  the   time  from  the   date 
of  discount  to  the  maturity  of  the  note. 


Baiik  Discount  231 

570.  To  find  the  Bank  Discount  and  Proceeds,  when  the  Face 
of  a  note,  Rate,  and  Time  are  given. 

1.  What  is  the  bank  discount  of  $368  for  3  mo.  at  6^? 
What  are  the  proceeds  ? 

Solution. — The  face  of  the  note  =     $368 

Int.  of  $1  for  3  mo.  and  grace  at  6%     =    .0155 

Discount  —  $5,704 

Proceeds,  $368- $5. 704  =  $363,296.     Hence,  the 

EuLE. — Find  the  interest  of  the  note  at  the  given  rate 
for  three  days  more  than  the  specified  time ;  the  result 
is  the  discount. 

Subtract  the  discount  from  the  face  of  the  ?iote;  the 
remainder  will  be  the  proceeds. 

Note. — If  a  note  is  on  interest,  find  its  amount  at  maturity,  and 
taking  this  as  the  face  of  the  note,  cast  the  interest  on  it  as  above. 


• 


2.  Find  the  proceeds  of  a  note  of  S650,  due  in  3  mo.,  at  6%, 

3.  Find  the  proceeds  of  a  draft  of  $825,  on  60  days,  at  6%. 

4.  Find  the  maturity  and  term  of  discount  of  a  note  of 
$1250,  at  5%  int.,  on  60  days,  dated  July  1st,  1880,  and  dis- 
counted Aug.  21st,  1880,  at  6%.     What  were  the  proceeds? 

5.  Find  the  difference  between  the  true  and  bank  discount 
on  $4000  for  1  year,  allowing  each  3  days  grace,  at  7%  ? 

6.  A  merchant  bought  $6500  worth  of  goods  for  cash,  sold 
them  on  4  months,  at  15^  advance,  and  got  the  note  dis- 
counted at  6^c  to  pay  the  bill.     How  much  did  he  make  ? 

571.  To  find  the  Face  of  a  note,  when  the  Proceeds,  Rate,  and 

Time  are  given. 

1.  For  what  sum  must  a  note  be  made  on  4  months,  that 
the  proceeds  may  be  $640,  discounted  at  6^  ? 

Solution.— The  bank  discount  of  $1  for  4  mo.  3  d.  =:  $.0205 
The  proceeds  of  $1  =  $l-$.0205  =  $.9795 

Therefore,  The  face  of  the  note  is  $640 ^$.9795       =$653,394 

Hence,  the 

Rule. — Divide  the  given  proceeds  by  the  proceeds  of  $1 
for  the  given  time  and  rate. 


232  Percentage. 

2.  What  must  be  the  face  of  a  note  on  6  months,  discounted 
at  1%,  that  the  proceeds  may  be  1500  ? 

3.  The  avails  of  a  note  were  $4350.90,  the  term  4  months, 
and  the  rate  of  discount  8^  ;  what  was  the  face  of  the  note  ? 

4.  How  large  a  note  on  3  months,  must  I  have  discounted  at 
6^,  to  reaUze  15260  ready  money  ? 

Commercial    Discount. 

572.  Commercial  Discount  is  a  per  cent  deducted  from  the 
face  of  bills,  the  list  price  of  goods,  etc. 

573.  The  Net  Price  of  goods  is  the  sum  received  for  them. 

574.  To  find  Coniiiiercial  Discount,  when  the  rate  is  given. 

1.  What  is  the  commercial  discount  on  goods,  the  list  price 
of  which  is  1235,  sold  at  b%  off  ? 

Solution.— 5%  is  .05,  and  $335 x. 05  =  $11.75,  Ans. 

2.  What  was  the  net  price  received  for  a  parlor  organ,  whose 
list  price  was  $450  on  3  mo.,  at  7^^  off  for  cash  ? 

3.  What  is  the  net  value  of  a  bill  of  books,  amounting  to 
$568.50,  on  60  days,  at  10;^  off  for  cash  ? 

4.  After  0%  had  been  deducted  from  the  list  price,  a  bill 
of  goods  was  sold  for  $625  ;  what  was  the  list  price  ? 

5.  Sold  a  bill  of  goods  amounting  to'  $850,  on  4  mo.,  at  8^ 
discount,  and  deducted  b%  for  cash ;  what  was  the  net  price  ? 

Solution.— 1850  x  .08  =  $68.     And  $850 -$68  =  $782. 

Again,  $782  x  .05  =  $39.10,  and  $782-$39.10  =  $742.90.     Hence,  the 

EuLE. — Deduct  the  discount  from  the   marhed  price, 
and  from  the  remainder  tahe  the  discount  for  cash. 

6.  What  is  the  net  value  of  a  bill  of  goods  amounting  to 
$2560,  sold  at  10$  discount  and  4$  off  for  cash  ? 

7.  What  is  the  net  value  of  a  cargo  of  flour  invoiced  at 
$3765,  at  12$  discount  and  5$  off  for  cash  ? 

8.  Find  the  net  value  of  a  bill  amounting  to  $4372,  at  15$ 
discount  and  2|$  off  for  cash  ? 


Commercial  Discount.  233 

9.  Find  the  sum  received  for  a  sale  of  goods  marked  at 
$6500,  at  S%  discount  and  ^1%  off  for  cash  ? 

10.  What  is  the  cash  value  of  a  bill  of  $10000,  at  7^  discount 
and  '\:^%  off  for  cash  ? 

11.  Find  the  net  value  of  the  following  :  63  lb.  tea,  at  88  cts., 
sold  on  3  mo.,  8^  off ;  95  boxes  of  starch,  at  68  cts.,  4^  off ; 
54  drums  of  figs,  at  75  cts.,  ^c  off;  85  bbl.  flour,  at  $7.50,  10% 
off  ;  allowing  ^%  discount  for  cash. 

575.  To   Mark  goods  so  that  a  given  per  cent   may  be  deducted 
and  leave  a  given  per  cent  profit. 

1.  Bought  ladies'  hats  at  15.10 ;  what  price  must  they  be 
marked,  that  15^  may  be  deducted  and  leave  20^  profit  ? 

Analysis. —The  selling  price  is  120%  of  $5.10,  and  $5.10  x  1.20  =$6.13. 
But  the  marked  price  is  to  be  diminished  by  15%  of  itself,  and  100%  — 
15%  =  85%  ;  hence,  $6.12  is  85%  of  the  marked  price.  Now  .|6.12  ^  .85 
=  $7.20,  the  marked  price,     (Art.  466.)     Hence,  the 

KuLE. — Find  the  selling  -price  and  divide  it  hy  1  minus 
the  given  per  cent  to  he  deducted;  the  quotient  will  be 
the  marhed  price. 

2.  Paid  %oQ  for  a  sewing  machine ;  what  must  I  ask  for  it 
that  I  may  abate  5^  and  sell  it  at  a  gain  of  25%  ? 

3.  A  shoe  dealer  paid  $3.60  a  pair  for  boots;  what  must  he 
ask  for  them  that  he  may  deduct  1%'^%  and  make  16f^  ? 

4.  A  jeweller  bought  diamond  rings  at  1120 ;  what  must  he 
ask  for  them  that  he  may  abate  4^  and  still  make  20^  ? 

5.  Bought  a  piano  for  $250  ;  what  must  I  ask  for  it  that  I 
may  deduct  20^  and  leave  a  profit  of  20^  ? 

Questions. 

563.  What  is  discount  ?  564.  True  discount  ?  The  present  worth  of  a 
debt?    565.  How  found? 

566.  What  is  bank  discount  ?  567.  The  proceeds  of  a  note  ?  568.  When 
does  a  note  mature  ?    569.  Wliat  is  the  term  of  disconxit  ? 

570.  How  find  the  discount  of  a  note  ?  The  proceeds  ?  571.  How  find 
the  face  of  a  note  that  the  proceeds  may  be  a  given  sum  ?  572.  What  is 
commercial  discount  ?    573.  What  is  the  net  price  of  goods  ? 


W'         /  .\fc    .       ..  ln»1i>,         ,.,      r;Txv^^-P 


iQUATION  OF    (PAYMENTS. 


DevetjOip^ient    of    I^rinciples. 

576.  1.  How  long  must  $1  be  kept  on  interest  to  equal  the 
interest  of  $2  for  3  mo.  at  tlie  same  rate  per  cent  ? 

Analysis. — As  $2  are  twice  $1,  at  the  same  rate  $1  must  be  kept  on 
interest  twice  as  long  as  $2,  and  2  times  3  mo.  are  6  months,  Ans. 

2.  How  long  must  12  be  kept  on  interest  to  equal  the  interest 
of  18  for  3  months  ? 

3.  How  long  must  13  be  kept  on  interest  to  balance  the 
interest  of  $9  for  4  months  ? 

4.  How  long  must  110  be  kept  to  balance  the  interest  of  $5 
for  4  months  ? 

Analysis.— |10  is  twice  $5 ;  therefore,  $10  must  be  kept  half  as  long 
as  $5,  and  ^  of  4  mo.  is  3  months,  Ans. 

5.  How  long  will  it  take  130  to  balance  the  interest  of  110 
for  6  months  ? 

6.  In  what  time  will  the  interest  of  1200  balance  the  interest 
of  $50  for  8  months  ? 

577.  From  the  examples  above  we  deriye  the  following 

Principles. 

1°.   The  rate  and  time  remaining  the  same, 
Double  the  principal  producer  twice  the  interest. 
Half  the  ^jrincipal  produces  half  the  i^iterest,  etc. 

2°.  The  rate  and  principal  remaining  the  same. 
Double  the  time  produces  twice  the  interest. 
Half  the  time  produces  half  the  interest,  etc.     Hence, 

578.  The  interest  of  any  given  principal  for  1  year,  1  month, 
or  1  day,  is  the  same  as  the  interest  of  1  dollar  for  as  many  years, 
months,  or  days,  as  there  are  dollars  in  the  given  principal. 


Equation  of  Payments,  ^35 

579.  Eq[uation  of  Payments  is  the  method  of  finding  the 
average  time  for  the  payment  of  several  debts,  due  at  different 
times,  without  loss  of  interest  to  either  party. 

580.  The  Average  or  Eq[uated  time  is  the  date  when  the 
several  payments  m.ay  be  made  at  one  time. 

581.  The  Term  of  Credit  is  the  time  before  a  debt  becomes 
due. 

582.  The  Average  Term  of  Credit  is  the  time  at  which  debts 
due  at  different  times  may  be  equitably  paid. 

Written     Exercises. 

583.  To  find  the  Averdge  Tune,  when  the  items  have  the  same 

date,  but  different  terms  of  credit. 

1.  Bought  Oct.  10th,  1880,  the  following  bills  of  goods,  for 
which  I  was  to  pay  $485  cash,  $200  in  2  mo. ;  $275  in  4  mo. ; 
and  $360  in  5  mo.;  what  is  the  average  time  and  the  date, 
when  these  bills  may  be  paid  without  loss  to  either  party? 

ExPLANATiox.— The  first  bill  is  cash  and 
has  no  interest.  The  int.  of  |200  for  3  mo. 
is  the  same  as  the  int.  of  $1  for  400  mo. 
(Prin.  r)  The  int.  of  $275  for  4  mo.  is  the 
same  as  that  of  $1  for  1100  mo.  The  int.  of 
$300  for  5  mo.  is  the  same  as  that  of  $1  for 
1800  months.  Therefore,  the  amount  of 
interest  due  on  the  whole  debt,  is  equal  to 
the  interest  on  $1  for  3800  mo.  Now  as  $1  is  entitled  to  int.  for  3300 
months,  the  whole  debt  $1320  is  entitled  to  interest  for  ygV^  of  3800  mo., 
and  8300  -f- 1320  =  2^  months,  the  average  term  of  credit. 

And  21  mo  added  to  Oct.  10th,  1880  =  Dec.  25th,  1880,  the  date  of 
payment.     Hence,  the 

Rule. — Multiply  each  item  hy  its  term  of  credit,  and 
divide  tlie  sum  of  the  products  hy  the  sum  of  the  items. 
The  quotient  luilt  he  the  average  term  of  credit. 

Adding  the  average  term  of  credit  to  the  date  of  the 
Bill,  will  give  the  date  of  payment. 

Notes. — 1.  When  an  item  contains  cents,  if  less  than  50,  they  are 
rejected  ;  if  50  or  more,  $1  is  added 


OPEKATION. 

$485     X    0     = 

0 

200  X  2  = 

400 

275  X  4  m 

1100 

360  X  5  = 

1800 

$1320 

3300 

236  Percentage, 

2.  In  the  quotient,  a  fraction  less  than  |  d.,  is  rejected  ;  if  4  d.  or  more, 
1  day  is  added. 

2.  A  merchant  buys  goods,  and  agrees  to  pay  1400  down, 
$400  in  4  months,  and  1400  in  8  months  ;  what  is  the  average 
time  of  the  whole  ? 

3.  A  man  borrows  1600,  and  agrees  to  pay  $100  in  2  months, 
$200  in  5  months,  and  the  balance  in  8  months;  when  can  he 
justly  pay  the  whole  at  once? 

4.  A  man  buys  a  house  for  $1600,  and  agrees  to  pay  $400 
down,  and  the  rest  in  3  equal  annual  instalments ;  what  is  the 
average  term  of  credit  ? 

5.  I  have  $1200  owing  to  me,  \  of  which  is  now  due ;  |-  of  it 
will  be  due  in  4  montlis,  and  the  remainder  in  8  months ;  what 
is  the  average  term  of  credit  ? 

6.  A  grocer  bought  goods  amounting  to  $1500,  for  which  he 
was  to  pay  $250  down,  $300  in  4  months,  and  $950  in  9 
months ;  when  may  he  pay  the  whole  at  once  ? 

7.  A  young  man  bought  a  farm  for  $2000,  and  agrees  to  pay 
$500  down,  and  the  balance  in  5  equal  annual  instalments; 
what  is  the  average  term  of  credit  ? 

584.  To  find  the  Average  Time,  when  the  terms  of  credit  are 
different,  and  begin  at  different  dates. 

8.  Bought  goods  as  follows  :  March  1st,  1880,  $200  on  2  mo.; 
April  6th,  $800  on  4  mo.;  June  17th,  $1000  on  3  mo.;  what  is 
the  average  time  and  date  of  payment  ? 

OPERATION. 

$200  due  May   1,  00  d.  x  200  =  00 

800  "  Aug.   6,  97  d.  X  800  =   77600 

1000  "  Sept.  17,  139  d.  x  ^000  =  139000 

2000    )  216600 

The  average  time  is  108  d.     (Art.  583,  N.)     108 
Date  of  payment  108  d.  from  May  1st,  or  Aug.  17th,  1880. 

Explanation. — Taking  as  the  standard  the  earliest  date  at  which 
either  of  the  items  becomes  due  (May  Istj,  the  term  of  credit  to  Aug.  6,  is 
97  d.,  to  Sept.  17tli,  139  days.  The  average  term  of  credit  is  therefore 
108  days,  and  the  date  of  payment  is  Aug.  17th,  1880.     Hence,  the 


Equation  of  Payments.  237 

Rule. — I.  Find  the  date  when  each  item  matures. 
Take  the  first  day  of  the  month  in  which  the  earliest 
item  becomes  due  as  a  standard,  and  find  the  number  of 
days  from  this  to  the  maturity  of  each  of  the  other  items. 

II.  Multiply  each  item  by  its  number  of  days,  and 
divide  the  sum  of  the  products  by  the  sum  of  the  items 
The  quotient  will  be  the  average  term  of  credit. 

III.  Add  the  average  time  to  the  standard  date,  and 
the  result  will  be  the  equitable  date  of  payment. 

Note. — Any  date  may  be  assumed  as  the  standard,  but  it  is  most 
convenient  to  take  the  first  day  of  the  month  in  which  the  earliest  item 
falls  due. 

9.  Bought  the  following  amount  of  goods  on  4  months' 
credit  :  March  10th,  1879,  $200 ;  April  15th,  1160  ;  May  1st, 
$440;  at  what  time  is  the  amount  payable  ? 

10.  Bought  the  following  bills  on  8  months  :  July  5th,  1879, 
1620.25  ;  Aug.  11th,  $240.56  ;  Sept.  20th,  $321.64 ;  Oct.  12th, 
$510.38;  Nov.  1st,  $308.17  ;  when  ought  a  note  for  the  whole 
amount  to  be  dated  ? 

11.  A  merchant  bouglit  the  following  bills  of  goods  :  March 
19th,  $350  on  4  mo.;  April  1st,  $430  on  130  days;  May  16th, 
$540  on  95  days ;  June  10th,  $730  on  3  mo. ;  what  is  the 
average  time  for  payment  of  the  whole  ? 

12.  Bought  the  followiug  bills  of  goods  on  90  days'  credit : 
May  10th,  $375.63;  May  18th,  $738.45;  June  3d,  $860.40; 
June  17th,  $692.38  ;  July  3d,  $379.68  ;  July  12th,  $417.13;  at 
what  time  will  the  whole  be  due  at  once  ? 

13.  A  grocer  sold  the  following  amount  of  goods  :  June  3d; 
$380  on  90  days'  credit  ;  June  10th,  $485  on  30  d. ;  July 
21st,  $834  on  60  d.  ;  July  27th,  $573  on  110  d. ;  Aag.  2d, 
$485  on  80  d. ;  when  will  the  whole  be  due  ? 

14.  Sold  the  following  bills  of  goods  on  3  months ;  Sept. 
5th,  1880,  $1163.25;  Sept.  20th,  $2368.41  ;  Oct.  7th,  $3561.34; 
Oct.  23d,  $840.90;  Nov.  13th,  $1307.63;  at  what  time  must 
a  note  for  the  whole  amount  be  dated  to  give  the  buyer  the 
specified  credit  ? 


338 


Percentage. 


Averaging-    Accounts. 

585.  Averaging  an  Account  is  finding  tlie  equated  time  at 
which  the  balance  may  be  joaid. 

586.  To  find  the  Average  Tune  for  settling  an  account. 

1.  Find  the  equated  time  and  date  of  paying  the  balance  of 
the  following  account : 

Dr,      John  Hamiltoi^  in  acct.  with  Henry  Morgan.      Cr. 


1881. 

\         1881. 

Jan.        5 

For  Mdse.  2  mo. 

1300 

Jan.      25 

By  Draft  90  d. 

1200 

Feb.      26 

"        "      3  mo. 

200 

March  28 

''  Cash. 

300 

March  28 

"      1  mo. 

500 

May      25 

"  Cash. 

100 

OPERATION. 


Due. 
Maicli     5 

Amt. 
$300 

Time. 
4d. 

Prod. 
1200 

Due. 
Apr.      28 

Amt. 
$200 

Time. 
58  d. 

Prod. 
11600 

Mav      26 

200 

86 

17200 

March  28 

300 

27 

8100 

April    28 

500 

58 

29000 

May      25 

100 

85 

8500 

$1000 
600 

Bal.     $400 


47400 

28200 


600 


28200 


)  19200  (  48  days. 


Ans.  Bal.  $400,  due  in  48  days  from  March  1st,  or  April  18th. 


Explanation. — Having  found  when  each  item  of  debt  and  credit 
becomes  due,  by  adding  its  term  of  credit  to  its  date,  we  assume  as  the 
standard  date  the  first  day  of  the  month  in  wliich  the  earliest  item  on  either 
side  of  the  account  matures,  viz.:  March  1st. 

Multiply  each  item  on  both  sides  by  the  number  of  days  between  the 
standard  date  and  the  maturity  of  each  item,  and  divide  the  difference 
between  the  sums  of  the  products  (19200),  by  the  difference  between  the 
sums  of  the  items  (400).     The  quotient  is  the  average  time  of  payment. 

Since  the  equated  time  requires  the  interest  of  $1  for  19200  days,  it  will 
require  $400,  ^^^  part  as  long,  and  19200-^400.=  48 ;  and  48  days  added 
to  March  1st  gives  April  18th.     Hence,  the 


Averaging  Accounts. 


239 


Rule. — I.  Write^  the  date  at  which  each  item  on  both 
sides  matures,  and  assume  the  first  day  of  the  montli  in 
which  the  earliest  item  on  either  side  becomes  due,  as 
the  standard  date.  Find  the  number  of  days  from  this 
standard  to  the  matuj^ity  of  the  respective  items. 
(Art.  583,  N.) 

II.  Multiply  each  item  by  its  Jiumber  of  days,  and 
divide  the  difference  between  the  sums  of  products  by  the 
difference  between  the  sums  of  items ;  the  quotient  will 
be  the  average  time. 

III.  //  tJte  grea,ter  sum  of  items  and  the  greater  sum 
of  products  are  both  on  the  same  side,  add  the  average 
time  to  the  assumed  date;  if  on  opposite  sides,  subtract 
it ;  and  the  result  will  be  the  date  when  the  balance  of 
the  account  is  equitably  due. 

Notes. — 1.  In  finding  the  maturity  of  notes  and  drafts,  3  days  grace 
should  be  added  to  the  specified  time  of  payment. 

3.  When  no  time  of  credit  is  mentioned,  the  transaction  is  understood 
to  be  for  cash,  and  its  payment  due  at  once. 

2.  Find  the  average  time  of  papng  the  following  account  : 
Dr.  George  Hadley.  Or. 


1880. 

1880. 

March  1 

To  Mdse. 

1500 

Apr.  12 

By  Draft,  20  d. 

$300 

Apr.     5 

"       "         2  mo. 

700 

May   10 

'-'   Cash. 

540 

May  20 

"       "        4  mo. 

650 

June    4 

a         a 

500 

3.  At  what  date  can  the  balance  of  the  followins:  account  be 


equitably  paid  ? 
Dr. 


W.  H.  Hendersoi^^, 


Cr. 


1881. 

1881. 

Apr.     7 

To  Mdse.,  2  mo. 

$300 

May     1 

To  Mdse., 

60  d. 

1350 

July     5 

"        1  mo. 

500 

June  10 

a          a 

30  d. 

500 

Aug.  10 

"       "        1  mo. 

400 

Aug.  30 

"   Cash. 

200 

240 


P&rcentage. 


4.  Average  the  following  account : 
Dr.  James  Brown  &  Co. 


Cr 


1882. 

1882. 

Jan.   10 

To  Mdse.,  3  mo. 

$400 

Jan.     1 

By  Bal.  of  Acct. 

$485 

"     25 

"       "        30  d. 

265 

Feb.  10 

"    Note,  3  mo. 

2500 

Apr.  20 

"       "        2  mo. 

850 

March  1 

"    Draft,  30  d. 

260 

5.  Balance  the  following  account  : 
Dr,  C.  J.  Hammoi^d. 


CV. 


1880. 

1880. 

Jan.  20 

To  Sundries 

,30d. 

$500 

Jan.    20 

By  real  estate  60  d. 

$400 

Feb.  12 

a            a 

60  d. 

340 

March  1 

"  Draft         60  d. 

200 

March  1 

a            i( 

30  d. 

300 

"    20 

"  Cash. 

400 

6.  Average  the  following  account : 
Dr.  He:n^ry  Eatmond  &  Co. 


Or 


1881. 

1881. 

Aug.  10 

To  Mdse.,  60  d. 

$150 

x\ug.   25 

By  Mdse., 

30  d. 

$500 

Oct.      1 

'^    Cash. 

350 

Sept.  20 

a            a 

20  d. 

300 

''      18 

"    Dft.       30  d. 

200 

7.  Find  when  the  balance  of  the  following  account  becomes 
due  : 

A.  B.  bought  of  C.  D.,  July  16th,  1882,  merchandise  $350 ; 
Aug.  11th,  $460  ;  Sept.  9th,  $570;  Sept.  14th,  $840;  Oct.  18th, 
$780.  The  former  paid  August  1st,  $260  ;  Sept.  30th,  in  grain 
$340  ;  Oct.  5th,  cash  $500 ;  Oct.  21st,  $625. 

Questions. 

577.  When  the  time  and  rate  of  interest  remain  the  same,  what  is  the 
effect  of  doubling  the  principal  ?  The  principal  and  rate  remaining-  the 
same,  what  is  the  effect  of  doubling  the  time  ? 

579.  What  is  equation  of  payments?  580.  What  is  average  or  equated 
time?  581.  The  term  of  credit  ?  586.  Describe  the  process  of  averaging 
accounts. 


* 


T  O  C  K  S . 


Definitions 


587.  A  Corporation  is  a  company  authorized  by  law  to 
transact  business  as  a  single  individual,  having  the  same  rights 
and  ohligations. 

588.  Stock  is  the  Capital  or  money  used  by  a  corporation  in 
carrying  on  its  business. 

589.  A  Share  is  one  of  the  equal  parts  into  which  the  stock 
is  divided. 

Note. — The  'oalue  of  a  share  varies  in  different  companies.  It  is  usu- 
ally $100,  and  will  be  so  regarded  in  this  work,  unless  otherwise  stated. 

590.  A  Certificate  of  Stock  is  a  written  instrument  issued  by 
a  corporation,  stating  the  number  of  shares  to  which  the 
holder  is  entitled,  and  the  original  value  of  each  share. 

591.  The  Par  Value  of  stock  is  the  sum  named  in  the 
certificate. 

592.  The  Market  Value  is  the  sum  for  wiiich  it  sells. 

Notes. — 1.  When  shares  sell  for  their  nominal  value,  they  are  a.t  par  ; 
when  they  sell  for  more,  they  are  above  par,  or  at  a  premium  ;  when  they 
sell  for  less,  they  are  below  par,  or  at  a  discount. 

2.  When  stocks  sell  at  par  they  are  often  quoted  at  100  ;  when  at  7% 
above  par,  they  are  quoted  at  107,  or  at  7  5?^  premium  ;  when  at  15%  below 
par,  they  are  quoted  at  85,  or  at  15%  discount 

593.  An  Assessment  is  a  percentage  required  of  stockholders 
to  replace  losses,  etc. 

594.  The  Gross  Earnings  of  a  company  are  its  entire 
receipts. 

11 


242  Percentage. 

595.  The  Net  Earnings  are  the  remainder  after  all  expenses 
are  deducted. 

596.  A  Dividend  is  a  percentage  divided  among  the 
stock-holders. 

597.  A  Bond  is  a  written  agreement  to  pay  a  sum  of  money 
at  or  before  a  specified  time. 

Notes. — 1.  U.  S.  Bonds  are  generally  designated  according  to  the  rates 
of  interest  tliey  bear.  Thus,  U.  S.  5's  denote  bonds  issued  by  the  United 
States  bearing  5%  interest ;  U.  S.  4's,  those  bearing  4%,  etc. 

2.  Bonds  of  States,  cities,  corporations,  etc.,  are  named  by  combining 
the  rate  of  interest  they  bear  with  the  name  of  the  State,  corporation,  etc., 
by  which  they  are  issued  j  as,  Ohio  6's,  N.  Y.  Central  5's,  etc. 

598.  A  Coupon  is  a  certifica.te  of  interest  due  on  a  bond,  to 
be  cut  off  when  paid,  as  a  receipt. 

599.  The  term  Stocks  is  applied  to  government,  state,  city, 
and  railroad  bonds,  to  the  capital,  of  banks,  etc. 

600.  Premiums,  discounts,  dividends,  and  assessments  are 
calculated  by  Percentage. 

The  jpar  value  of  the  stock  is  the  hase  ;  the  'per  cent  of  pre- 
mium or  discount  is  the  rate  ;  the  premium  or  discount  is  the 
percentage;  the  par  value  phis  the  premium  i^  ^t\\Q  amount ; 
and  the  par  value  minus  the  discount  is  the  difference. 

Written     Exercises. 

601.  To  find  the  Premium,  Discount,  Dividend,  etc.,  from  the  Par 

Value  and  Rate. 

Formula. — Premium,  etc.  ==  Par  Value  x  Rate. 

1.  What  is  the  premium,  at  7^,  on  40  shares  of  bank  stock  ? 

2.  What  is  the  discount,  at  15  5"^,  on  50  shares  of  railroad 
stock  ? 

3.  What  is  the  dividend,  at  6^,  on  85  shares  of  telegraph 
stock  ? 

4.  Find  the  assessment,  at  10%,  on  42  shares  of  oil  stock  ? 


Stocks.  243 

602.  To  find  the  Market  Value  of  Stock  from  the  Par  Value 

and  the  Premium  or  Discount. 

^  ^^    ,  ^  ^^  ^  (  Par  Value  +  Premium. 

Formula. — Marlcet  Value  =   '  „      rr  7  r^- 

(  Far  value  —  Discount. 

5.  Required  the  market  value  of  23  shares  of  bank  stock,  at 
7^  premium  ? 

6.  Find  the  market  value  of  28  shares  of  telegraph  stock,  at 
7^  discount  ? 

7.  What  cost  87  shares  of  iron  mountain  stock,  at  \h%  pre- 
mium and  brokerage  \%  ? 

8.  Find  the  cost  of  150  shares  of  insurance  stock,  at  '^\%  dis- 
count, brokerage  \%  ? 

9.  What  is  the  cost  of  100  shares  of  N.  Y.  and  New  Haven 
R.  R.  stock,  at  125,  brokerage  \%  ? 

Oral     Exercises. 

603.  1.  A  premium  of  120  was  paid  on  4  shares  of  bank 

stock  ;  what  was  the  rate  per  cent  ? 

Analysis. — Since  4  sliares  pay  $20,  one  share  ($100)  pays  \  of  $20,  or 
$5.     Therefore,  the  rate  was  jf^ ,  or  5  % . 


2.  Bought  10  shares  of  stock  for  which  a  premium  of  %{ 
was  paid  ;  what  was  the  rate  of  premium  ? 

3.  Paid  a  premium  of  150  on  20  shares  of  oil  stock ;  what 
was  the  rate  per  cent  ? 

4.  Sold  15  shares  of  mining  stock  for  $75  ;  what  was  the 
rate  of  discount  ? 

Written     Exercises. 

604.  To    find    the    Rate    from    the    Par    Value,    the     Premium, 
Discount,  Dividend,  etc. 

1.  The  gross  receipts  of  a  manufacturing  company  are 
$17250,  the  expenses  are  $6250,  and  its  capital  $50000 ;  what 
per  cent  dividend  can  it  make  ? 


244  Percentage. 

Analysis.— The  receipts  less  expenses  are  $17250  — $6250  =  $11000. 
Now  as  $50000  are  entitled  to  $11000,  $1  is  entitled  to  $1 1000  h- $50000 

=  .22,  or  22%.     Hence,  the 

^  T,  J         i  Freynium,  Discount,  ]        „      ^^  , 

FOKMULA.— i^fif^fe  =  \         r>--77^  \  ^  P^L'T    VolUG. 

(      Dividend,  etc.  \ 

2.  A  premium  of  $375  was  paid  for  25  shares  of  E.  E.  stock ; 
at  what  rate  was  the  premium  ? 

3.  The  discount  on  50  shares  of  the  Pacific  Eaih-oad  was 
$625  ;  what  was  the  rate  of  the  discount  ? 

4.  If  the  income  on  $2356  is  $268.50,  what  is  the  rate  %  ? 

5.  What  per  cent  of  3648  acres  is  456  acres  ? 

605.  To  find  the  Cost  of  a  given  number  of  shares,  the  market 
value  of  one  share  and  the  rate  of  brokerage  being  given. 

6.  What  cost  15  shares  of  E.  E.  stock,  at  120,  brokerage  \%  ? 

Analysis.— The  cost  of  1  share  is  129%  +i%  brokerage,  or  120^%  of 
$100  =  $120.25,  and  15  shares  will  cost  $1803.75,  Ans.    Hence,  the 

-r^  ^    ,        i  Market  Value  of  1  sliare -\- Brolceraqe 

Formula. — Cod  =  ^  ,^     ^        ;   ,  ^ 

{      X  Number  of  shares. 

7.  What  is  the  cost  of  78  shares  of  E.  E.  stock  at  124},  and 
brokerage  at  \%  ? 

8.  Find  the  cost  of  121  shares,  at  89|^,  and  brokerage  |^? 

9.  Sold  250  shares  of  bank  stock  at  87f ,  and  paid  ^%  broker- 
age ;  how  much  did  I  receive  for  it  ? 

10.  What  is  the  cost  of  375  shares  of  National  Express  stock, 
at  25^  premium  and  brokerage  -|%  ? 

606.  To  find  the  Number  of  Shares,  when  the  investment  and 

the  cost  of  I  share  are  given. 

11.  How  many  shares  of  bank  stock  at  h%  discount  and 
brokerage  1%,  can  be  bought  for  $7620  ? 

Analysis. — Since  the  discount  is  5  %  and  brokerage  \  % ,  the  cost  of  1 
share  is  95 %+\%,ot  95^ %  of  $100  -  $95.25.  As  $95.25  will  buy  1  share, 
$7620  will  buy  as  many  shares  as  $95  25  are  contained  times  in  $7620, 
and  $7620 -^$95.25  =  80  shares,  Ans.     Hence,  the 

FoKMULA. — Numher  of  Shares = luvestmenf  ~  Cost  of  1  Share. 


Stocks.  245 

12.  How  many  shares  of  telegraph  stocky  at  7|^  premium 
and  brokerage  1%,  can  you  buy  for  $13500? 

13.  Fmd  the  number  of  shares  of  mining  stock  at  102|,  that 
can  be  bought  for  $5150,  and  brokerage  ^%. 

14.  What  number  of  raih'oad  shares  at  125,  brokerage  i%, 
will  $7515  pay  for  ? 

15.  How  many  shares  of  express  stock,  at  10^  premium, 
can  be  bought  for  18030  ? 

16.  Find  the  number  of  shares,  at  20^  discount,  that  can  be 
bought  for  $3200  ? 

607.  To  find  how  stock  must  be  bought  which  pays  a  given  pep 
cent  dividend,  to  realize  a  given  per  cent  on  the  investment. 

17.  At  what  price  must  I  bny  Western  K.  R.  stock  which 
pays  6^  dividend,  so  as  to  realize  8^o  on  the  investment? 

Analysis. — Dividend  .06-^.08  income  =  .75,  or  75%,  price  of  stock. 
FoEMULA. — Price  =  Dividend  -^  Rate  of  Income. 

18.  What  must  be  paid  for  4^  bonds  that  the  investment 
may  yield  Q%^ 

19.  What  must  be  paid  for  U.  S.  5's  that  8^  may  be  receiyed 
on  the  investment  ? 

20.  W^hat  must  be  paid  for  stock  that  yields  10^  dividends, 
so  as  to  realize  lY/o  on  the  investment  ? 

608.  To  find  what  sum  to  invest  to  yield  a  given  income,  the  cost 
of  I  share,  rate  of  interest,  or  dividend  being  given. 

21.  What  sum  must  be  invested  in  N.  Y.  5's,  at  108-J-,  to 
produce  an  annual  income  of  $1500  ? 

Analysis. — The  income  $1500-^|o  (int.  on  1  share)  =  300  shares,  and 
108|  (price  of  1  share)  x  300  =  $32550.     Hence,  the 

FoEMULA. — Investment  =  Cost  1  Share  x  Number  of  Shares. 

22.  What  sum  must  be  invested  in  U.  S.  4's,  at  105,  to  yield 
$3000  annually  ? 

23.  What  sum  must  be  invested  in  Nebraska  8's,  at  75,  to 
yield  an  income  of  $1540  annually  ? 


246  Percentage. 

24.  Wbat  sum  must  be  invested  in  stock  at  112,  which  pays 
V)%  annually,  to  obtain  an  income  of  12200  ? 

25.  What  sum  must  be  invested  in  Alabama  6's,  at  85,  to 
realize  $2000  a  year  ? 

26.  How  much  must  be  invested  in  stock  at  106,  to  yield  an 
income  of  1600,  the  stock  paying  10^  dividend  annually  ? 

609.  To  find  the  rate  per  cent  of  income  from    bonds  paying  a 
given  rate  of  interest,  and   bought  at  a  given   premium  or  discount 
without  regard  to  their  maturity. 

27.  What  is  the  rate  of  income  on  bonds  paying  8^  interest; 
bought  at  112  ? 

Solution.— Interest  on  1  share  $8-r-112,  cost  per  share  =  7^:^,  Ans, 

28.  Bought  bonds  paying  6^  interest,  at  75 ;  what  was  the 
rate  per  cent  of  income  ? 

Solution. — Interest  of  1  share  |6-5-75  cost  per  share  =  8%,  Ans. 

Interest  per  Share 


Formula. — Bate  %  Income  —  ,        ^    , 

[  -^  (Jost  per  bfiare. 

29.  Find  the  per  cent  of  income  on  U.  S.  5's,  bought 
at  110. 

30.  What  is  the  per  cent  of  income  on  Iowa  6's,  bought  at 
108,  brokerage  \%  ? 

31.  Which  is  the  more  profitable,  $10000  invested  in  3|-  per 
cents  at  75,  or  in  7  per  cents  at  105  ? 

610.  To  find   the  rate  per  cent  of  income  from  bonds  paying  a 
given  rate  of  interest,  bought  at    a  given  premium  or  discount  and 
payable  at  par  in  a  given  time. 

32.  What  rate  per  cent  income  will  be  realized  from  N.  Y. 
5's,  bought  at  a  premium  of  8^,  and  paid  at  par  in  10  years  ? 

Analysis. — Since  the  bond  matures  in  10  years,  the  premium  on  1  share 
($8)  decreases  -fj^,  or  ||  each  year.  Now  the  interest  $5— $|==$41,  annual 
income  on  1  share.  And  $4|^  -i-  108,  cost  of  1  share  =  .OS^,  or  3y  %,  the 
rate  required. 


Stocks.  247 

33.  What  rate  per  cent  income  will  be  realized  from  North 
Carolina  8's,  bought  at  90,  if  paid  at  par  in  20  years  ? 

Analysis. — Since  the  bond  matures  in  20  years,  the  average  decrease 
of  the  discount  on  1  share  is  $10  -i-  $20  =  $|  each  year.  Now  the  interest 
$8  +  $i  =  |8|,  the  annual  income  on  1  share.  And  $8.50  -4-  $90  (cost  of 
1  share)  =  $.09|,  or  9|%,  the  rate  required.     Hence,  the 

KuLE. — First  find  the  average  annual  decrease  of  the 
premiiun  or  discount. 

If  the  bonds  are  a,t  a  premium,  subtract  it  from  the 
given  rate  of  interest ;  if  at  a  discount,  add  it  to  the 
interest;  the  result  will  he  the  average  income  of  one 
share. 

Divide  the  average  income  of  one  share  by  the  cost  of 
one  share,  and  the  quotient  will  be  the  rate  per  cent  of 
income. 

Notes. — 1.  When  bonds  are  at  a  'premium,  the  longer  the  time  before 
maturity,  the  greater  will  be  the  rate  per  cent  of  income. 

2.  When  bonds  are  at  a  discount,  the  longer  the  time  before  maturity, 
the  less  will  be  the  rate  per  cent  of  income. 

34.  What  rate  per  cent  of  income  will  be  received  on  U.  S. 
4's  at  106,  and  payable  at  par  in  15  years  ? 

35.  Bought  Milwaukee  and  St.  Paul  bonds  at  90,  due  at  par 
in  30  years,  drawing  10^  interest ;  what  is  the  rate  per  cent  of 
income  ? 

Q  U  ESTION  S. 

587.  What  is  a  corporation  ?  588.  Stock  ?  589.  A  share  ?  590.  A  cer- 
tificate of  stock  ?     591.  The  par  value  ?    592.  Market  value  ? 

593.  An  assessment  ?  594.  Gross  earnings  ?  595.  Net  earnings?  596. 
A  dividend  ?    597.  A  bond  V    598.  A  coupon  ? 

599.  To  what  is  the  term  stocks  applied  ?  600.  How  are  premiums, 
etc. ,  computed  ? 

601.  What  does  the  premium  equal?  602.  The  market  value?  604. 
The  rate?  605.  The  cost?  606.  The  number  of  shares?  607.  How 
find  the  price  ?  609.  How  find  the  rate  of  income  without  regard  to 
maturity  ?    610.  On  bonds  payable  at  par  at  maturity  ? 


IXCHANGE. 


611.  Exchange  is  a  method  of  making  payments  between 
distant  places  without  sending  the  money. 

612.  A  Draft  or  Bill  of  Exchange  is  a  written  order  direct- 
ing one  j^erson  to  pay  another  a  certain  sum,  at  a  specified  time. 

613.  The  Drawer  is  the  person  who  signs  the  draft. 

614.  The  Drawee  is  the  joerson  to  whom  it  is  addressed. 

615.  The  Payee  is  the  person  to  whom  the  money  is  to 
be  paid. 

616.  A  Sight  Draft  is  one  payable  on  its  idvesentation, 

617  A  Time  Draft  is  one  payable  at  a  specified  time  after 
date  or  presentation. 

Note. — Drafts  or  Bills  of  Exchange  are  negotiable  like  promissory 
notes,  and  the  laws  respecting  them  are  essentially  the  same. 

618.  An  Acceptance  of  a  draft  is  an  engagement  to  pay  it. 
As  evidence,  the  drawee  writes  the  word  accepted  across  the 
face  of  the  draft,  with  the  date  and  his  name. 

619.  The  Par  of  Exchange  is  the  standard  by  which  the 
Talue  of  the  currency  of  different  countries  is  compared,  and  is 
either  intrinsic  or  commercial. 

620.  Intrinsic  Par  is  a  standard  having  a  reed  and  fixed 
value  represented  by  gold  or  silver  coin. 

621.  Commercial  Par  is  a  conventional  standard,  having  any 
assumed  value  which  convenience  may  suggest. 

Note. — The  fluctuation  in  the  price  of  bills  from  their  par  value,  is 
called  the  Course  of  Exchange. 


Domestic  Exchange,  249 


Domestic    Exchange. 

622.  Domestic  Exchange  is  a  method  of  making  payments 
between  distant  places  in  the  same  conntry. 

623.  To  find  the  Cost  of  a  Draft,  when   the   Face  and   Rate  of 

Exchange  are  given. 

1.    What  cost  the  following  aiglit  draft,  at  '%\%  premium  ? 


$2500. 


New  York,  June  30tli,  1881. 


At  sight,  pay  to  the  order  of  James  Clark,  twenty-five 
hundred  dollars,  vsilue  received,  and  charge  the  same  to  the 
account  of 


To  S.  Bareett  &  Co., 
New  Orleans,  La. 


Smith  Bros.,  &  Co. 


Analysis.  —  Since    exchange    on  operation. 

N.  O.   is   2i%  prera.,  the  cost  of  $1  1  +  .025  =r  $1,025 

draft  is  ^1.025,  and  $2500  will   cost  $1,025   X  2500  =  $2562.50 
$1,025  X  2500  =  $2562.50,  Ans. 

2.  What  is  the  cost  of  a  sight  draft  on  San  Francisco  for 

13000,  at  2i%  discount  ? 

Solution.— A  draft  of  $1  at  $2|%  discount  will  cost  $0,975,  and  $3000 
X  .975  =  $2925.00,  Ans.     Hence,  the 

Rule. — Multiply  the  face  of  the  draft  by  the  cost  of  $1. 

624.  On  time  drafts,  both  the  rate  of  exchange  and  bank 
discount  are  commonly  included  in  the  rate,  which  in  quotations 
for  time  drafts  is  enough  less  than  for  sight  drafts,  to  allow  for 
bank  discount. 

Required  the  cost  of  a  sight  draft 

3.  On  St.  Louis,  at  1^%  premium,  for  $850  ? 

4.  On  Buffalo,  at  \%  discount,  for  $975  ? 

5.  On  Savannah,  Ga.,  for  $2000,  at  1|^  premium  ? 


250  Percentage. 

6.  What  is  the  cost  of  the  following  time  drafts  at  \^%  pre- 
mium, and  interest  at  6^  ? 

^Jfi^O^  "  Philadelphia,  Julj  5tli,  1881. 

Sixty  days  after  sight,  pay  to  the  order  of  George 
Wilcox,  four  thousand  dollars,  value  received,  and  charge  the 
same  to  the  account  of 


H.  Adams  &  Co. 


To  S.  Parkhurst, 
Trenton,  N.  J. 


Analysis. — At  \\%  premium,  tlie  opeeation. 

cost  of  $1  draft   at   sifflit  is  $1,015.  %1  -f  $0,015  —  $1,015 

But  the  draft  is  subject  to   interest  ^    ^  $.0105  =  $0.0105 

for  60  d.  +  8  d.  grace.     The  int.  of  n         ^      ^     .e 

$1  for  63  d.,  at  6%  is  $0.0105,    and  ^^^^  ^1  <^^^ft  =  $1.0045 
$1,015-0.0105  =  $1.0045  the  cost  of        $1.0045   X  4000  =  $4018 
$1  draft,   and  a  draft  of  $4000  will 
cost  4000  times  $1.0045,  or  $4018,  Ans. 

7.  Find  the  cost  in  Omaha  of  a  draft  on  New  York  at  90 
days  sight,  for  $5265,  at  %%  premium,  interest  being  Q\%. 

8.  Required  the  worth  in  Memphis  of  a  draft  on  Boston  for 
$3500,  at  30  days  sight,  at  1%  discount  and  interest  Q>%. 

9.  What  is  the  worth  of  a  draft  of  $5000  on  St.  Louis, 
at  30  days  sight,  premium  1^%,  including  interest  ? 

625.  To  find  the  Face  of  a  Draft,  when  the  Cost  and  Rate  of 
Exchange  are  given. 

10.  How  large  a  draft  on  Philadelphia  can  he  bought  in 
Charleston,  at  60  days  sight,  for  $3000,  the  premium  being 
lY/c,  and  interest  6^  ? 

Analysis  — Since  the  premium  is  opekatton. 

li%,   the  cost  of  $1     sight    draft  $1  -|-  .015  =  $1,015 

would  be  $1,015.      But  the  bank  Bank  dis.  63  d.  =       .0105 

discount   on    $1    for   63  d.  is  .0105,  f  a,^  a     «-    _  ^^ 

hence,  the  cost  of  $1  draft  is  $1.0045.  ^^'^^  ^^  ^^  <il''^^-  —  '^1-0045 

Now  if  $1  draft  cost  $1.0045,  $8000  $3000  -f-  1.0045  =  $2986.56 
will  buy  a  draft  of  as  many  dollars 
as  $1.0045  is  contained  times  in  $3000,  or  $2986.56,  An8.     Hence,  the 


Foreign  Exchange.  251 

Rule. — Divide  the  cost  of  the  draft  by  the  cost  of  $1 
exchange. 

11.  What  was  the  face  of  a  sight  draft  for  which  12500  was 
paid,  exchange  being  at  "1^%  premium  ? 

12.  What  is  the  face  of  a  sight  draft  bought  for  $3300  in 
Memphis  on  Boston,  exchange  being  ?)\%  discount  ? 

13.  What  is  the  face  of  a  draft  on  4  months  for  which  $450 
is  paid,  exchange  \%  premium,  and  int.  6^  ? 

14.  Find  the  face  of  a  draft  on  New  York  at  90  days  sight, 
bought  in  Cincinnati  for  $2250,  exchange  at  \\%  discount, 
interest  h%  ? 

15.  A  merchant  of  Galveston  paid  $4265  for  a  draft  on  St. 
Louis  at  30  d.  sight,  exchange  at  2>\%  premium,  interest  %% ; 
what  was  the  face  of  the  draft  ? 

16.  What  is  the  face  of  a  draft  on  Cincinnati  at  90  days 
sight,  bought  for  $3000,  exchange  2^%  premium,  interest  Q%  ? 


Foreign    Exchange. 

626.  Foreign  Exchange  is  the  method  of  making  payments 
between  different  countries. 

627.  A  Set  of  Exchange  consists  of  three  bills  of  the  same 
date  and  tenor,  First,  Second,  and  Third  of  exchange.  They 
are  sent  by  different  mails  in  order  to  save  time  in  case  of  mis- 
carriage.    W^hen  one  is  paid,  the  others  are  void. 

628.  Exchange  with  Europe  is  chiefly  done  through  large 
commercial  centers,  as  London,  Paris,  Geneva,  Amsterdam, 
Antwerp,  Hamburg,  Frankfort,  and  Berlin. 

629.  Bills  drawn  on  England,  Scotland,  or  Ireland,  are 
called  Sterling  Bills,  and  the  value  of  a  Pound  Sterling  is 
quoted  in  U.  S.  money. 

630.  The  present  Par  of  Exchange  on  Great  Britain  is 
$4.8665  gold  to  the  pound  sterling,  which  is  the  intrinsic  value 
of  a  Sovereign,  as  estimated  at  the  U.  S.  Mint. 


252 


Percentage. 


631.  In  quoting  exchange  on  a  foreign  country,  it  is  cus- 
tomary to  quote  the  value  of  the  money  unit  of  that  country  in 
U.  S.  money. 

Note. — These  values  are  published  annually  by  the  Secretary  of  th© 
Treasury.     Those  given  on  the  1  st  day  of  Jan. ,  ]  882,  are  as  follows  : 


Country. 

Austria 

Belgium 

Bolivia 

Brazil 

British  N.  A 

Chili 

Cuba 

Denmark 

Ecuador  

Egypt 

France 

Great  Britain.  .  .  . 

Greece 

German  Empire.. 

Hay  ti 

India 

Italy 

Japan 

Liberia 

Mexico 

Netherlands 

Norway 

Peru 

Portugal 

Kussia 

Sandwich  Islands 

Spain 

Sweden 

Switzerland 

Tripoli 

Turkey 

U.  S.  of  Colombia 
Venezuela 


Monetary  Unit. 

Florin 

Franc 

Boliviano 

Milreisof  1000  reis... 

Dollar 

Peso 

Peso , 

Crown 

Peso 

Piaster „ . 

Franc 

Pound  sterling 

Drachma 

Mark 

Gourde 

Rupee  of  16  annas. . .  . 

Lira 

Yen 

Dollar 

Dollar 

Florin 

Crown 

Sol 

Mil  reis  of  1000  reis. . . 
Rouble  of  100  copecks 

Dollar 

Peseta  of  100  centimes 

Crown 

Franc 

Mahbub  of  20  piasters 

Piaster  

Peso 

Bolivar 


Standard. 


Silver 

Gold  and  silver.. 

Silver 

Gold 

Gold 

Gold  and  silver,. 
Gold  and  silver.. 

Gold 

Silver 

Gold 

Gold  and  silver.. 

Gold 

Gold  and  silver. . 

Gold 

Gold  and  silver. . 

Silver 

Gold  and  silver. . 

Silver 

Gold 

Silver  

Gold  and  silver. . , 

Gold 

Silver 

Gold  

Silver 

Gold 

Gold  and  silver. . 

Gold 

Gold  and  silver. .. 

Silver 

Gold 

Silver 

Gold  and  silver.  . 


U 


Value  in 
.  S.  Money. 


.40,7 
.19,3 
.82,3 
.54,6 
p.  00 
.91,2 
.93,2 
.26,8 
.82,3 
.04,9 
.19,3 

4.86.61 
.19,3 
.23,8 
.96,5 
.89 
.19,3 
.88,8 

1.00 
.89,4 
.40,2 
.26,8 
.82,3 

1.08 
.65,8 

1.00 
.19,3 
.26,8 
.19,3 
.74,3 
.04,4 
.82,3 
.19,3 


Foreign  Exchange.  253 

632.  The  method  of   finding  the  cost   of  foreign   bills  is 
essentially  the  same  as  that  of  domestic  bills.     (Art.  624. ) 

0 

1.  What  is  the  cost  of  the  following  bill  on  London,  at 
$4.8665  to  the  £  sterling  ? 


£35 Jf  12s.  Ne^  York,  July  4th,  1880. 

At  sight  of  tliis  first  of  exchange  (the  second  and  third 
of  the  sayne  date  and  tenor  unpaid),  pay  to  the  order  of 
Henry  Crosby,  three  hundred  fifty  four  pounds,  twelve 
shillings  sterling,  value  received,  and  charge  the  sa?ne  to  the 
account  of 

J.  Kii^G  &  Co. 

To  Geokge  Peabody,  Esq.,  London. 

Analysis.— £354  12s.  =  £354.6.    (Art.  403.)    As  4.8665 

£1  is  worth  $4.8665,  £354.6  are  worth  354.6  times  as  354.6 

much  ;  and  $4.8665  x  354.6  ■-=  $1725,661,  the  cost.  j^^^^^  $1725  661 

2.  What  is  the  cost  of  a  bill  on  Liverpool  for  £345  5s.  6d., 

at  $4,875  to  the  pound  sterling? 

3.  What  is  the  cost  in  currency  of  a  bill  on  Edinburgh  for 
£360.5,  exchange  being  at  par  and  gold  6%  premium  ? 

633.  Bills  on  Paris,  Anttverp,  and  Geneva,  are  quoted  by 
the  number  of  francs  and  centimes  to  a  dollar  in  gold. 

Note. — Centimes  are  commonly  written  as  decimals  of  a  Franc. 

4.  What  is  the  cost  of  a  bill  on  Paris  for  575  francs,  at  5.16 
francs  to  a  dollar  in  gold  ? 

Analysis. — As  5.16  francs  cost  $1,  575  francs  will  cost  as  many  dollars 
as  5.16  is  contained  times  in  575,  and  575  -^  5.16  =  $111.43,  Ans. 

5.  Find  the  cost  of  a  bill  on  Geneva  for  750.25  francs,  at 
5.15-|-  fr.  to  the  dollar  in  gold. 

6.  Find  the  cost  of  a  bill  for  1000  francs  on  Antwerp,  at 
5.17J  fr.  to  a  dollar,  gold  at  1<}(  premium  ? 


354  Percentage. 

634.  Bills  on  Bremen,  Frankfort,  Hamburg,  and  Berlin, 
are  quoted  by  the  value  in  U.  S.  Money  of  four  marks  (reichs- 
marks)  in  gold.  % 

7.  What  cost  a  bill  on  Frankfort  for  540  marks,  at  |.  94|-  ? 

Analysis. — Since  4  marks  are  worth  $.945.  the  worth  of  540  marks  is 
540  times  i  of  $.945,  or  $127.58,  Ans.     (Art.  632.) 

8.  What  cost  a  bill  on  Berlin  for  2800  marks  at  1. 96 J  in  gold  ? 

635.  The  method  of  finding  the  face  of  a  foreign  bill  of 
exchange  is  essentially  the  same  as  that  of  domestic  bills. 

9.  What  is  the  face  of  a  bill  of  exchange  on  London, 
bought  for  $4500  at  $4.87^  in  gold  ? 

Analysis.— Since  $4,875  will  buy  a  bill  of  £1,  $4500  will  buy  as  many 
pounds  as  $4,875  are  contained  times  in  $4500,  and  $4500  -j-  4.875  = 
£923.076,  or  £923  Is.  G^d.,  Ans. 

10.  What  is  the  face  of  a  bill  on  Dublin  for  which  $6500 
was  paid  in  gold,  at  $4.86  ? 

11.  What  is  the  face  of  a  bill  on  Paris  for  $2400,  exchange 
being  5.15  fr.  to  a  dollar  ? 

12.  Find  the  face  of  a  bill  on  Geneva,  which  cost  $1500 
gold,  exchange  5.16. 

13.  Find  the  face  of  a  bill  on  Frankfort  costing  $762  in 
gold,  exchange  at  95J. 

14.  Paid  $2000  for  a  bill  on  Berlin,  exchange  93f ;  what  was 
the  amount  of  the  bill  ? 


Qu  ESTI  ONS. 

611.  What  is  Exchange  ?  612.  A  draft  or  bill  of  exchange  ?  613.  The 
drawer  of  a  bill?  614.  The  drawee?  615.  The  payee?  616.  A  sight 
draft  or  bill  ?  617.  A  time  draft  or  bill  ?  618.  What  is  the  acceptance  of 
a  draft  ?    619.  The  par  of  exchange  ? 

622.  What  ia domestic  exchange?  623.  How  find  the  cost  of  a  draft? 
625.  How  find  the  face  ? 

626.  What  is  foreign  exchange  ?  627.  A  set  of  exchange  ?  620.  What 
are  sterling  bills?  631.  How  is  foreign  exchange  quoted?  632.  How 
find  the  cost  of  foreign  bills  ?    635.  How  find  the  face  ? 


^- •        4   h^ 


636.  Business  men  have  a  method  of  solving  practical  ques- 
tions, which  is  frequently  shorter  and  more  expeditious^  than 
that  of  arithmeticians  fresh  from  the  schools. 

637.  Their  method  consists  in  Analysis,  and  may,  with 
propriety,  be  called  the  Common  Sense  Method. 

638.  No  specific  directions  can  be  given  for  the  analysis  of 
problems.  The  learner  must  be  taught  to  depend  on  his 
judgynent  as  a  guide. 

639.  He  may,  however,  be  aided  by  the  following : 

General     Principles. 

1°.  We  reason  from  that  which  is  self-evident,  or  Tcnoimi,  to 
that  which  is  unknown,  or  required. 

2°.  We  reason  from  a  part  to  the  whole  j  as,  iohe7i  the  value 
of  one  is  given,  and  the  value  oftiuo  or  more  of  the  same  hind  is 
required. 

S°.  We  reason  from  the  whole  to  a  part  j  as,  when  the  value 
of  two  or  more  is  given,  and  that  of  a  part  is  required. 

Ji°.  We  reason  from  a  given  cause  to  its  effect,  or  from  agfven 
effect  to  its  cause ;  as,  when  different  comhinations  of  numhers 
are  given,  to  find  the  result. 

Thus,  If  3  men  can  mow  6  acres  in  1  day,  how  many  acres  can  they 
mow  in  5  days  ? 

Or,  If  to  draw  4  tons  requires  6  horses,  how  many  horses  will  be 
required  to  draw  8  tons  ? 


256  General  Analysis. 


Oral    Exer  cises. 

640.  1.  If  8  tons  of  coal  cost  140,  how  much,  will  6  tons 
cost? 

Analysis. — 1  is  |  of  8;  therefore,  1  ton  will  cost  1  eighth  of  $40, 
which  is  $5.     As  1  ton  costs  $5,  6  tons  will  cost  6  times  $5,  or  $30,  Ans. 

Or,  thus  :  6  tons  are  f  of  8  tons  ;  therefore,  6  tons  will  cost  f  of  $40 
Now  1  eighth  of  $40  is  $5,  and  6  eighths  are  6  times  $5,  or  $30,  the  same 
as  before. 

2.  If  7  lb.  of  tea  cost  42  shillings,  what  will  10  lb.  cost 

3.  If  9  sheep  are  worth  $27,  how  much  are  15  sheep  worth  ? 

4.  If  10  barrels  of  flour  cost  $60,  what  will  12  barrels  cost  ? 

5.  If  a  man  walks  54  kilometers  in  6  days,  how  far  does  he 
walk  in  15  days  ? 

6.  If  12  men  can  build  48  rods  of  wall  in  a  day,  how  many 
rods  can  20  men  build  in  the  same  time  ? 

7.  Suppose  75  kilos  of  butter  last  a  family  25  days,  how 
many  kilos  will  supply  them  12  days? 

8.  If  7  meters  of  cloth  cost  $30,  how  mucli  will  9  meters 
cost  ? 

9.  If  10  bbl.  of  beef  cost  $72,  how  much  will  8  bbl.  cost  ? 

10.  If  7  acres  of  land  cost  $50,  what  will  12  acres  cost? 

11.  If  f  ton  coal  cost  $6,  what  will  5  tons  cost? 

Analysis. — Since  3  fourths  ton  cost  $6,  1  fourth  will  cost  \  of  $6,  or 
|2,  and  4  fourths,  or  1  ton  will  cost  4  times  $2,  or  $8.  Now  at  $8  a  ton, 
5  tons  will  cost  5  times  $8,  or  $40,  Ans. 

12.  If  f  lb.  tea  cost  40  cts.,  what  will  12  lb.  cost  ? 

13.  If  5 J  bbl.  apples  cost  $22,  what  will  9  bbl.  cost  ? 

14.  If  -J  acre  land  cost  $28,  what  will  10  acres  cost  ? 

15.  If  16  cords  of  wood  are  worth  $48,  how  much  is  |  cord 
worth  ? 

16.  If  I  of  a  citron  cost  28  cents,  what  must  you  pay  for 
12  citrons  ? 

17.  If  I  yd.  cloth  cost  $2,  what  is  that  a  yard  ? 

18.  If  4  lb.  ginger  cost  $f ,  what  will  11  lb.  cost  ? 

19.  If  3  melons  cost  $yV,  what  will  20  melons  cost  ? 

20.  Paid  $^0  for  4  slates  ;  what  must  I  pay  for  18  slates  ? 


General  Analysis,  257 


Written    Exercises. 

641.  The  pupil  should  be  required  to  Analyze  each  of  the 
following  examples,  giving  results  as  he  proceeds,  and  be 
encouraged  to  invent  different  solutions. 

1.  If  60  bbl.  flour  cost  $300,  what  will  42  bbl.  cost  ? 

Analysis. — Since  60  barrels  cost  $300,  1  barrel  costs  gV  ^^  $300,  and 
$300  4- 60  =  $5.00. 

Again,  42  barrels  will  cost  42  times  as  mncli  as  1  barrel.     Therefore, 

$5  X  42  =  $210,  Alls. 

Note. — Mucli  labor  may  often  be  saved  by  indicating  the  operations 
required,  and  cancelling  the  common  factors  before  the  multiplications 
and  divisions  are  performed. 

5 

Thus,  %^-§-  X  42  =  %^-  X  42  =  S210,  Ans, 

2.  A  man  bought  30  cords  of  wood  for  $76.80 ;  how  much 
must  he  pay  for  195  cords  ? 

3.  A  gentleman  bought  85  meters  of  carpeting  for  $106.25  ; 
how  much  would  38  meters  cost  ? 

4.  A  drover  bought  350  sheep  for  $525 ;  how  much  would 
65  cost,  at  the  same  rate  ? 

5.  If  12|^  pounds  of  coffee  cost  $1.25,  how  much  will  245 
pounds  cost  ? 

6.  If  126  bushels  of  corn  are  worth  $52.92,  how  much  are  84 
bushels  worth  ? 

7.  Paid  $20  for  60  pounds  of  tea  ;  how  much  would  112| 
pounds  cost,  at  the  same  rate  ? 

8.  Bought  41  meters  of  flannel  for  $16.40;  how  much  would 
18|-  meters  cost  ? 

9.  Bought  18  pounds  of  ginger  for  $4.50 ;  how  much  will 
20|  pounds  cost? 

10.  If  a  stage  goes  84  miles  in  12  hours,  how  far  will  it  go  in 
108  hours  ? 

11.  If  16  horses  eat  72  bushels  of  oats  in  a  week,  how  many 
bushels  will  125  horses  eat  in  the  same  time  ? 

12.  If  a  railroad  car  goes  120  miles  in  5  hours,  how  far  will 
it  go  in  212J  hours  ? 


358  General  Analysis, 

13.  If  a  steamboat  goes  189  kilometers  in  12  hours,  how  far 
will  it  go  in  5|  hours  ? 

14.  If  -^  of  a  cord  of  wood  costs  f  of  a  dollar,  how  much 
will  }  of  a  cord  cost  ? 

ANA1.TSIS. — Since  y\  cord  costs  $|,  ^^  cord  will  cost  \  of  $|,  or  $|-,  and 
}f  or  1  cord  will  cost,  $^  x  13  =  %--i~,  or  1|.  Again,  since  1  cord  costs  $\^-, 
I  cord  will  cost  |  of  $V-  =  fi  or  $li. 

Or,  $1  X-1/-XI  zz:  Ifx^Xf  =  If  nr  %1\,    AuS. 

15.  If  f  of  a  yard  of  cloth  cost  £-f,  how  much  will  |  of  a 
yard  cost  ? 

16.  If  -^  of  a  ship  cost  116000,  how  much  is  |  of  her  worth  ? 

17.  If  a  man  pays  $47  for  building  23|-  rods  of  fence,  how 
much  would  it  cost  him  to  build  213f  rods  ? 

18.  A  farmer  paid  $45.42  for  building  36|  rods  of  stone  wall ; 
how  much  will  it  cost  him  to  build  QO-^  rods  ? 

19.  If  7|-  meters  of  satinet  cost  $9|,  how  much  will  18J 
meters  cost? 

20.  A  ship's  company  of  30  men  have  4500  pounds  of  flour  ; 
how  long  will  it  last  them,  allowing  each  man  2J  lb.  per  day  ? 

21.  How  long  will  56700  pounds  of  meat  last  a  garrison  of 
756  men,  allowing  1^  lb.  apiece  per  day  ? 

22.  A  can  chop  a  cord  of  wood  in  4  hours,  and  B  in  6  hours  ; 
how  long  will  it  take  both  to  chop  a  cord  ? 

Analysis. — Since  A  can  chop  a  cord  in  4  hours,  in  1  hour  he  can  chop 
^  of  a  cord ;  and  since  B  can  chop  a  cord  in  6  hr.,  he  can  chop  |^  of  a  cord  ; 
hence  both  can  chop  ^  +  ^  cord  =  fW  cord  in  1  hr. 

Again,  if  to  chop  /^  cord  takes  both  1  hr.,  to  chop  J^  cord  will  take  I 
hour,  and  ||  or  a  whole  cord  will  take  them  12  times  i  hr.,  or  2|  hours. 

23.  If  a  man  can  plant  a  field  in  8  days,  and  a  boy  in  12 
days  ;  how  long  will  it  take  both  to  plant  it  ? 

24.  A  can  do  a  piece  of  work  in  20,  B  in  40,  and  C  in  60 
minutes  ;  how  long  would  it  take  all  together  to  do  it  ? 

25.  A  cistern  has  3  faucets,  one  of  which  will  empty  it  in 
5  hr.,  another  in  10  hr.,  and  the  other  in  15  hr. ;  how  long  will 
it  take  all  3  to  empty  it  ? 


General  Analysis.  259 

26.  If  10  men  require  8f  days  to  finish  a  piece  of  work,  how 
long  will  it  take  11  men  to  finish  it  ? 

27.  A  water-tank  has  3  pipes;  the  first  will  empty  it  in  12  hr., 
the  second  will  fill  it  in  6  hr.,  the  third  in  8  hours ;  in  how 
many  hours  will  the  tank  be  filled  if  all  run  together? 

28.  A  deer  starting  150  rods  before  a  dog,  runs  30  rods  a 
minute  ;  the  dog  follows  at  the  rate  of  42  rods  a  minute.  In 
what  time  will  the  dog  overtake  the  deer  ? 

Oral     Exercises. 

642.  1.  A  grocer  sold  8  lb.  of  sugar  at  12  cents  a  pound, 
and  took  his  pay  in  lard,  at  10  cents  a  pound  ;  how  much 
lard  did  it  take  to  pay  for  the  sugar  ? 

Analysis. — Since  1  lb.  of  sugar  is  worth  12  cents,  8  lb.  are  wortli  8 
times  12,  or  96  cents. 

Again,  since  10  cents  will  pay  for  1  lb.  of  lard,  96  cents  will  pay  for  as 
many  pounds  as  10  cents  are  contained  times  in  96  cents,  or  9|  pounds. 

Or,  thus  :  13  cents  are  jf  of  10  cents ;  therefore,  1  lb.  of  sugar  is  worth 
^  lb.  of  lard  ;  and  8  lb.  of  sugar  are  worth  8  times  {§  lb.  of  lard,  which 
is  fl  lb.  =  9i%  or  9|  lb.,  Ans. 

2.  How  many  dozen  eggs,  at  15  cents  a  dozen,  will  pay  for 
12^  yards  of  muslin,  worth  8  cents  a  yard  ? 

3.  A  farmer  exchanged  8  tons  of  hay,  worth  $20  per  ton, 
for  flour  worth  $6  a  barrel ;  how  many  barrels  of  flour  did  he 
receive  for  his  hay  ? 

4.  A  man  exchanged  50  lb.  of  wool,  valued  at  37^  cents  a 
pound,  for  flannel  worth  87J  cents  a  yard  ;  how  many  yards 
did  he  obtain  ? 

5.  A  lad  bought  75  apples,  at  the  rate  of  3  for  a  cent,  and 
exchanged  them  for  oranges  worth  5  cents  apiece  ;  how  many 
oranges  did  he  receive  ? 

6.  How  many  slate  pencils  worth  -J-  cent  apiece,  can  you 
buy  for  150  steel  pens  worth  4  cents  per  dozen  ? 

7.  How  many  acres  of  farming  land  worth  $12|-  per  acre, 
must  be  given  in  exchange  for  4  building  lots  in  the  city, 
valued  at  $75  per  lot  ? 


260  General  Analysis, 

8.  How  much  lard  at  27  cents  a  kilo,  will  pay  for  153  Kg.  of 
rice,  worth  9  cents  a  kilo  ? 

9.  How  many  oranges  at  7J  cents  apiece,  can  you  buy  for  \ 
of  35  quarts  of  strawberries,  at  Vl\  cents  a  quart  ? 

10.  A  lad  bought  12  peaches,  at  the  rate  of  3  for  4  cts.,  and 
afterwards  exchanged  them  for  oranges  which  were  3  for  8 
cents ;  how  many  oranges  did  he  obtain  ? 

11.  Frank  sold  10  apples,  which  was  f  of  all  he  had  ;  he 
then  divided  the  remainder  equally  among  5  companions;  how 
many  did  each  receive  ? 

12.  Lincoln  spent  60  cents  for  a  book,  which  was  \\  of  his 
money  ;  with  the  remainder  he  bought  oranges,  at  4  cents 
apiece  ;  how  many  oranges  did  he  buy  ? 

13.  A  man  paid  away  $35,  which  Avas  f  of  all  he  had ;  he 
then  spent  the  rest  in  cloth  at  12  per  yard  ;  how  many  yards 
did  he  obtain  ? 

14.  A  farmer  bought  a  quantity  of  goods,  and  paid  120  down, 
which  was  f  of  the  bill ;  how  many  cords  of  wood,  at  $3  per 
cord,  will  it  take  to  pay  the  balance  ? 

Written     Exercises. 

643.  1.  A  merchant  sold  75  yd.  silk,  at  10.84  a  yd.,  and  took 
his  pay  in  corn,  at  $0.60  a  bu.  ;  how  many  bu.  did  he  receive? 

Analysis.— At  $.84  a  yd.  75  yards  are  worth  $.84  x  75  =  63. 
Again,  to  pay  $63.00  will  require  as  many  bushels  of  corn  as  $.60  are 
contained  times  in  $63.00,  and  $63.00  -^  .60  =  105  bu.,  Ans. 

2.  How  many  pounds  of  butter,  at  35  cents,  must  be  given 
in  exchange  for  186^  yards  of  calico,  at  18J  cents  per  yard  ? 

3.  How  many  pounds  of  tobacco,  at  16J  cents,  must  be  given 
for  256  pounds  of  sugar,  at  6J  cents  a  pound? 

4.  A  farmer  bought  325  sheep  at  $2J-  apiece,  and  paid  for 
them,  in  hay,  at  $10|-  per  ton ;  how  many  tons  did  it  take  ? 

5.  A  man  bought  a  hogshead  of  vinegar,  worth  37-|-  cents 
per  gallon,  and  gave  331^  pounds  of  cheese  in  exchange ;  how 
much  was  the  cheese  a  pound  ? 


General  Analysis.  261 

6.  Bought  274  bushels  of  salt,  at  42^  cents  per  bushel,  and 
paid  in  wheat  at  $1.25  per  bushel ;  how  many  bushels  of  wheat 
did  it  require  ? 

7.  A  bookseller  exchanges  400  dictionaries  worth  $1.87|-  cents 
apiece,  for  900  grammars ;  how  much  did  the  grammars  cost 
apiece  ? 

8.  How  many  meters  of  silk,  worth  llf  per  meter,  will  pay 
for  249-|-  meters  of  cloth  worth  15:^  per  yard  ? 

9.  Bought  263|  yds.  of  satinet,  at  $lf  per  yard,  and  paid 
for  it  in  cheese,  at  $9-|-  per  hundred  ;  how  much  cheese  did 
it  take  ? 

10.  Bought  25  hhds.  22  gals.  3  qts.  of  molasses  at  37^  cents 
per  gallon,  and  paid  for  it  in  wool,  at  62|^  cents  a  pound  ;  how 
much  wool  did  it  take  ? 

11.  Bought  432  sheep  at  $2 J  apiece,  for  which  I  paid  144 
barrels  of  flour  ;  what  was  the  flour  per  barrel  ? 

12.  If  15  yards  of  flannel  are  worth  25  yards  of  muslin,  how 
many  yards  of  flannel  are  worth  315  yards  of  muslin  ? 

13.  A  market-woman  bought  10  dozen  oranges,  at  the  rate 
of  3  for  4  cents,  and  then  exchanged  them  for  eggs,  at  the  rate 
of  4  for  5  cents  ;  how  many  eggs  did  she  receive  ? 

14.  If  15  lbs.  of  pepper  are  worth  25  lbs.  of  ginger,  how 
many  pounds  of  ginger  must  be  given  for  195  lbs.  of  pepper? 

Oral     Exercises. 

644.     1.   What  cost  36  bushels  of  oats,  at  33|  cts.  a  bushel  ? 

Analysis. — At  $1  a  bushel  36  bu.  would  cost  $36 ;  but  the  price  is 
33|  cents,  or  %\  ;  lience,  at  %\,  36  bu.  will  cost  I  of  $36,  wliicli  is 
$12,  Ans.     (Art.  280.) 

2.  At  12|-  cts.  a  pound,  what  cost  72  lbs.  of  maple  sugar  ? 

3.  At  20  cts.  apiece,  what  will  150  melons  come  to  ? 

4.  What  cost  60  rolls  of  tape,  at  Q\  cts.  a  roll  ? 

5.  What  cost  72  yd.  delaine,  at  16f  cts.  a  yard  ? 

6.  At  25  cts.  a  yard,  what  must  I  pay  for  64  yards  of  ribbon  ? 

7.  At  50  cts.  a  bushel,  what  will  250  bushels  of  corn  cost  ? 

8.  At  33^  cts.  apiece,  what  cost  12  doz.  Grammars? 


262  General  Analysis, 

Written    Exercises. 

645.  lo  What  will  1268  bushels  of  apples  come  to,  at  25 

cts.  a  bushel  ? 

Analysis.— 25  cts.  =  %\  ;  therefore  the  apples  will  cost  \  as  many  dol- 
lars as  there  are  bushels,  and  1268-7-4  =  317.     Ans.  $317. 

2.  At  8  J  cts.  apiece,  what  cost  a  gross  of  slates  ? 

3.  What  cost  480  yards  of  ribbon,  at  16f  cts.  a  yard  ? 

4.  At  33-J  cts.  a  hektoliter,  what  must  I  pay  for  750  hekto- 
liters  of  potatoes  ? 

5.  What  cost  1250  melons,  at  20  cents  each  ? 

6.  At  50  cents  apiece,  how  much  will  1745  Readers  cost? 

Oral     Exercises. 

646.  1.  Two  boys  formed  a  partnership  in  selling  news- 
papers ;  A  put  in  30  cts.  and  B  50  cts.  They  gained  40  cts. 
the  first  day ;  what  was  the  share  of  each  ? 

Analysis. — Their  capital  was  30  cts.  +  50  cts.  =  80  cts. 
A's  part  of  it  was  f §,  or  | ;  and  B's  part  was  |§,  or  |. 
Now  I  of  40  is  5  cts.,  and  |  are  3  times  5,  or  15  cts.,  A's  share. 
And  f  are  5  times  5,  or  25  cts.,  B's  share. 

2.  A  and  B  bought  a  pony  together  for  $100 ;  A  put  in  $60 
and  B  $40  ;  they  sold  it  so  as  to  gain  130  ;  what  was  each  one's 
share  of  the  gain  ? 

3.  Two  men  buy  a  carriage  together  for  $500  ;  A  put  in 
$300  and  B  $200  ;  they  sold  it  at  a  loss  of  $150.  What  was  the 
share  of  each  in  the  loss  ? 

4.  C  and  D  joined  in  a  speculation  and  cleared  $90 ;  C  put 
in  1400  and  D  1800  ;  what  share  of  the  gain  had  each  ? 

5.  A  man  failed  in  business,  owing  A  $700  and  B  $400; 
his  property  was  valued  at  $880 ;  how  much  would  each 
creditor  receive  ? 

6.  B  and  C  engaged  in  business;  B  furnished  $900  and  C 
$600  ;  they  made  $300  ;  what  was  the  share  of  each  ? 


General  Analysis.  263 


Written     Exercises. 

647.  1.  A,  B,  and  0,  formed  a  partnership  ;  A  put  in 
$2000,  B  13000,  and  0  $4000  ;  they  gained  $2700 ;  what  was 
each  man's  share  of  the  gain  ? 

Ai^ALYSis.— The  capital  was  $2000  +  $3000  +  $4000  =  $9000. 
Since  A's  part  of  the  capital  was  |§^§,  or  |,  his  share  of  the  gain  was 
I  of  $2700,  and  f  of  |2700  =     $600    A's  gain. 

B's  part  was  f  g^g,  or  f  of  $2700,  and  |  of  $2700=     $900    B's  gain. 
C's  part  was  f§§§,  or  |  of  $2700,  and  f  of  $2700=  $1200    C's  gain. 

Proof,       $2700    Whole  gain. 

2.  A,  B,  and  C  hired  a  farm  together,  for  which  they  paid 
$175  rent ;  A  advanced  $75,  B  $60,  and  C  $40.  They  raised 
250  bushels  of  wheat ;  what  was  each  man's  share  ? 

3.  A,  B,  and  C  together  spent  $1000  in  mining  stocks.  A 
put  in  $400,  B  $250,  and  C  $350.  They  gained  $1500 ;  how 
much  was  each  man's  share  ? 

4.  A,  B,  C,  and  D  fitted  out  a  whale  ship ;  A  advanced 
$10000,  B  $12000,  0  $15000,  and  D  $8000.  The  ship  brought 
home  3000  bbls.  of  oil ;  what  was  each  man's  share  ? 

5.  A,  B,  and  0  formed  a  partnership  ;  A  furnished  $900, 
B  $1500,  and  0  $1200.  They  lost  $1260  ;  what  was  each  man's 
share  of  the  loss  ? 

6.  X,  Y,  and  Z  entered  into  a  joint  speculation,  on  a  capital 
of  $20000,  of  which  X  furnished  $5000,  Y  $7000,  and  Z  the 
balance.  Their  net  profits  were  $5000  per  annum  ;  what  was 
the  share  of  each  ? 

7.  A  bankrupt  owes  one  of  his  creditors  $300,  another  $400, 
and  a  third  $500.  His  property  amounts  to  $800  ;  how  much 
can  he  pay  on  a  dollar,  and  how^  much  will  each  of  his  creditors 
receive  ? 

8.  A  bankrupt  owes  $2000,  and  his  property  is  appraised  at 
$1600  ;  how  much  can  he  pay  on  a  dollar? 

9.  A  man  failing  in  business,  owes  A  $156.45,  B  $256.40, 
and  C  $360.40  ;  and  his  effects  are  valued  at  $317  ;  how  much 
will  each  man  receive  ? 


264  Gi^ueral  jLualyals. 

10.  The  assets  of  a  man  failing  in  business  amonnted  to 
$3560;  he  owed  $35600;  how  much  can  he  pay  on  a  dollar, 
and  how  much  will  B  receive,  who  has  a  claim  of  $5000  ? 

11.  A  man  died  insolvent,  owing  $55645,  and  his  property 
was  sold  at  auction  for  $2350 ;  how  much  will  his  estate  pay 
on  a  dollar  ? 

12.  A,  B,  and  C  sent  flour  by  sloop  from  New  York  to  Bos- 
ton. A  had  600  bbl.*,  B  400  bbl.,  and  C  200  bbl.  In  a  gale 
200  bbl.  were  thrown  overboard ;  what  was  the  loss  of  each  ? 

13.  A  and  B  formed  a  partnership  ;  A  put  in  $300  for  2 
months  and  B  $200  for  6  months.  They  gained  $150 ;  what 
was  each  man's  just  share  of  the  gain  ? 

Suggestion. — The  gain  of  each  depends  both  upon  the  capital  he  fur- 
nished, and  the  time  it  was  employed.     (Art.  583.) 

A's  capital  $300  x  3  =  $600,  the  same  as  $600  for  1  mo. 
B's        "        200  X  6  =  1200,        "        "     1200 

Whole  capital,  $1800 

A's  share  must  therefore  be  -^^^^  =  i  of  $150,  or  $50. 
B's        "        "        "        '*      i|^^  =  f  of  $150,  or  $100. 

Pkoof.— $50  +  $100  =  $150,  the  gain. 

14.  A,  B,  and  C  enter  into  partnership  ;  A  puts  in  $500  for 
4  mo.,  B  $400  for  6  mo.,  and  C  $800  for  3  mo. ;  they  gain 
$340  ;  what  is  each  man's  share  of  the  gain  ? 

15.  A  and  B  hire  a  pasture  together  for  $60 ;  A  put  in  120 
sheep  for  6  months,  and  B  put  in  180  sheep  for  4  months ; 
what  should  each  pay  ? 

16.  The  firm  A,  B,  and  C  lost  $246  ;  A  had  put  in  $85  for 
8  mo.,  B  $250  for  6  mo.,  and  C  $500  for  4  mo.;  what  is  each 
man's  share  of  the  loss  ? 

17.  Smith  and  Jones  graded  a  street  for  $857.50.  S.  fur- 
nished 5  men  for  20  days  and  6  men  for  15  days ;  J.  furnished 
10  men  for  12  days  and  9  men  for  20  days ;  what  was  the 
share  of  each  contractor  ? 

18.  Three  men  hire  a  farm  of  250  A.,  at  $8|-  an  acre ;  A  put 
in  244  sheep,  B  325,  and  C  450 ;  what  rent  ought  each  to 
pay? 


General  Analysis,  265 


Oral     Exercises. 

648.     1.  If  f  ton  of  hay  costs  115,  what  will  a  ton  cost  ? 

Analysis. — lu  this  example  we  have  a  'part  of  a  number  given,  to  find 
the  whole.  Since  15  is  f  of  the  number.  \  of  it  is  \  of  15,  which  is  5,  and 
I  are  4  times  5,  or  20.    Ans,  $20. 

Note. — In  solving  examples  of  this  kind  a  difficulty  often  arises  from 
supposing  that  if  f  of  a  certain  number  is  15,  \  of  it  must  be  \  of  15. 
This  mistake  will  be  easily  avoided  by  substituting  the  word  imrts  for  the 
given  denominator. 

Tlius,  if  3  parts  cost  $15,  1  part  will  cost  \  of  $15,  which  is  $5.  But 
this  part  is  a  fourth.  Now  if  1  fourth  cost  $5,  4  fourths  will  cost  4  times 
as  much. 

2.  A  builder  paid  120  for  |  of  an  acre  of  land ;  what  was 
that  per  acre  ? 

3.  A  boy  being  asked  how  many  pears  he  had,  replied  that 
he  had  50  apples,  which  was  f  the  number  of  his  pears  ;  how 
many  pears  had  he  ? 

4.  Henry  lost  42  yards  of  his  kite  line,  which  was  f  of  his 
whole  line  ;  what  was  its  length  ? 

5.  50  is  f  of  what  number  ?    |^  of  what  ?    -|  ?    y\  ? 

6.  75  is  J  of  what  number  ?    f  of  what  ?    -^  ?    -^\  ? 

7.  100  is  I  of  what  number  ?    |  of  what  ?     f  ?    H  ? 

8.  A  man  bought  a  yoke  of  oxen,  and  paid  156  in  cash, 
which  was  ^  of  the  price  of  them  ;  what  did  they  cost  ? 

9.  A  merchant  bought  a  quantity  of  wood  and  paid  $45  in 
goods,  which  was  f  of  the  whole  cost ;  how  much  did  he  pay 
for  the  wood  ? 

10.  A  man  bought  a  buggy  and  paid  S45  down,  which  was 
^  of  the  price  of  it ,  what  was  the  price,  and  how  much  did 
he  owe  ? 

11.  The  crew  of  a  whale  ship  having  been  out  24  months, 
found  they  had  consumed  ^  of  their  provisions ;  how  many 
months'  proyisions  had  they  when  they  embarked,  and  how 
much  longer  would  their  provisions  last  ? 

12.  A  man  bought  a  meadow  and  paid  $75  cash  which  was 
I  of  the  price,  and  gave  his  note  for  the  balance ;  how  large 
was  the  note  ? 


266  General  Aoialysis. 


Written     Exercises. 

649.  1.  A  man  being  asked  how  far  he  had  traveled, 
replied  that  140  miles  equaled  ^\  of  the  distance  ;  how  far  had 
he  traveled? 

Analysis.— Since  140  mi.  is  /g,  4^  is  f  of  140  mi.,  or  20  miles.  As  20 
mi.  is  aV  of  the  distance,  ||  is  25  times  20  mi.,  or  500  miles,  Ans. 

2.  560  is  f  J  of  what  number  ? 

3.  1500  is  I J  of  what  number  ? 

4.  2000  is  If  of  what  number  ? 

5.  A  man  paid  $150  for  a  carriage,  which  was  ff  of  what 
he  sold  it  for ;  what  did  he  get  for  his  carriage  ? 

6.  A  builder  paid  1145  for  -|  A.  of  laud  ;  what  was  that  an 
acre  ? 

7.  A  man  pays  $0.96  for  |  bu.  of  wheat,  what  is  that  a  bu.  ? 

8.  A  lad  being  asked  how  many  pears  he  had,  replied  that 
he  had  150  apples,  which  was  -^  the  number  of  his  pears ;  how 
many  pears  had  he  ? 

9.  680  is  I  of  what  number  ? 

10.  1260  is  If  of  what  number  ? 

11.  A  man  traveled  240  miles  by  railroad,  which  was  f  the 
distance  he  traveled  by  steamboat ;  how  far  did  he  go  by  boat  ? 

12.  If  4  times  |  of  $32  is  f  the  price  paid  for  a  cow,  what 
did  she  cost  ? 

13.  7  times  \  of  28,  is  ^  of  what  number  ? 

14.  7  times  f  of  36  cts.  is  ^  of  the  price  of  a  Dictionary  ; 
what  is  the  price  ? 

15.  A  man  spent  $560  for  books  which  was  |-§-  of  his  money, 
and  bought  hay  with  the  remainder,  at  $16  a  ton  ;  how  much 
hay  did  he  buy  ? 

16.  A  tailor  bought  a  horse  and  paid  $120  in  cash,  which  was 
^\  of  the  price  ;  how  many  coats  at  $24  apiece  will  it  take  to 
pay  the  balance  ? 

17.  A  grocer  sold  2205  lbs.  butter,  which  was  }|  of  all  he 
had  ;  how  many  tubs  would  hold  the  rem-ainder,  allowing 
42  lb.  to  a  tub  ? 


General  Analysis.  267 

18.  A  lad  being  asked  how  many   peaches  he  had  in  his 

basket,  replied  that  J,  J,  and  ^  of  them  made  104 ;  how  many 

had  he  ? 

Analysis.— The  sum  of  \,  a,  and  \  =  if.  (Art.  195.)  Now  if  104  is 
13^  ^^  is  _i_  of  104,  vvliicli  is  8  ;  and  [f  is  8  x  12  =  96.     Ans.  96  peaches. 

19.  A  farmer  lost  J  his  sheep  by  sickness,  \  by  wolves,  and 
he  had  72  sheep  left ;  how  many  had  he  at  first  ? 

20.  A  person  having  spent  \  and  -J  of  his  money,  finds  he 
has  148  left ;  what  had  he  at  first  ? 

21.  After  a  battle  a  general  found  that  \  of  his  army  had 
been  taken  prisoners,  \  were  killed,  -^^  had  deserted,  and  he 
had  900  left ;  how  many  had  he  before  the  battle. 

22.  A  certain  post  stands  -J  in  the  mud,  J  in  the  water,  and 
20  feet  above  the  water  ;  how  long  is  the  post  ? 

23.  Suppose  I  pay  r^l85  for  -|  of  an  acre  of  land ;  what  is 
that  per  acre  ? 

24.  A  man  paid  $2700  for  f^  of  a  vessel ;  what  is  the  whole 
vessel  worth  ? 

25.  A  gentleman  spent  ^  of  his  life  in  Boston,  J  of  it  in 
New  York,  and  the  rest  of  it,  which  was  30  years,  in  Philadel- 
phia ;  how  old  was  he  ? 

26.  AVhat  number  is  that,  -|  of  which  exceeds  -^  of  it  by  10? 

27.  In  a  school  \  were  studying  arithmetic,  \  algebra,  \ 
geometry,  and  the  remaining  18  were  studying  grammar  ;  how 
many  pupils  were  in  the  school  ? 

28.  A  owns  -J  and  B  -^ig  of  a  ship  ;  A's  part  is  worth  $650 
more  than  B's ;  what  is  the  value  of  the  ship  ? 

29.  In  a  certain  orchard  ^  are  apple  trees,  \  peach  trees, 
\  plum  trees,  and  the  remaining  15  were  cherry  trees  ;  how 
many  trees  did  the  orchard  contain  ? 

650.  1.  A  merchant  paid  $1165.25  for  a  case  of  goods  and 
sold  them  at  15^^  advance  ;  what  was  the  profit  ? 

Analysis.— The  profit  was  JjV  of  !^1165.25.  Now  ^  =  |11.6525,  and 
fl^  =  $11.6525  X  15  =  $174.7875,  Aus. 

2.  A  man  sold  a  house  for  $2969.50,  which  was  25^  more 
than  it  cost  him  ;  what  did  it  cost  him  ? 

Analysis. — Since  he  gained  35%,  he  received  $1.25  for  each  dollar  of 
coet.     Now  $2969. 50 -r- $1.25  =  $2375.60,  Ans. 


268  General  Analysis, 

3.  Eeceived  $4100  to  buy  stock,  after  deducting  2^%  com- 
mission ;  how  many  shares  at  par  can  I  buy  ? 

4.  What  is  the  premium  at  If^  for  insuring  $3560  on  a 
house  and  furniture  ? 

5.  What  sum  must  be  insured  on  goods  worth  $4760,  at 
^^%,  to  cover  both  the  goods  and  premium  ? 

6.  What  is  the  specific  duty  on  175  pieces  of  silk,  each  con^ 
taining  50  yd.,  at  25  cents  a  yard  ? 

7.  What  is  the  int.  of  $765.50  for  3  yr.  8  mo.,  at  Q%  ? 

8.  If  $850  at  simple  interest  amounts  to  $986  in  2  years, 
what  is  the  rate  per  cent  ? 

9.  When  money  is  at  6^,  what  is  the  present  worth  of 
$4218,  due  in  1  yi\  6  months? 

10.  What  is  the  bank  discount  on  a  note  of  $1640.50  for 
90  days,  at  b%  ? 

11.  What  must  be  the  face  of  a  note  on  60  d.,  at  %%,  to 
yield  $1000,  if  discounted  at  a  bank  ? 

12.  What  is  the  equated  time  for  the  payment  of  $400,  due 
in  3  mo.,  and  $600,  due  in  5  months  ? 

13.  In  a  mercantile  house,  A's  capital  is  $4500,  B's  $5200, 
and  C's  $5300  ;  they  make  $3000  ;  what  is  the  profit  of  each  ? 

14.  What  cost  a  sight  draft  on  New  Orleans  for  $750,  at  2f  ^ 
premium  ? 

15.  What  cost  a  bill  of  exchange  on  Liverpool  for  £800  10s., 
at$4.86i? 

651.  1.  A  father  divided  $2700  among  his  3  children  in 
the  proportion  of  2,  3,  and  4  ;  how  much  did  each  receive  ? 

Analysis.— The  sum  of  2  +  3  +  4  =  9.  Hence,  the  first  received  $2, 
the  second  $3,  and  the  third  .$4,  as  often  as  9  is  contained  in  $3700 ;  and 
9  is  contained  in  $2700,  300  times.  Therefore,  the  first  received  $800  x  2 
=  $600 ;  the  second  $300  x  3  =  $900  ;  the  third  $300  x  4  =  $1200. 

Pkoof.— $600  +  $900  +  $1200  =  $2700. 

2.  A  man  had  756  sheep  which  he  divided  into  2  flocks  in 
the  proportion  of  3  to  4  ;  how  many  were  in  each  flock  ? 


General  Analysis.  269 

3.  Divide  1248  into  2  sucli  parts  that  one  shall  be  to  the 
other  as  2  to  6. 

4.  Divide  435  into  2  such  parts  that  one  shall  be  3  times  the 
other. 

5.  Two  farmers  have  755  acres  of  land,  one  having  4  times 
as  many  acres  as  the  other  ;  hovt^  many  had  each  ? 

6.  Two  families,  one  containing  4  persons  the  other  5,  hired 
board  together  for  $2954  a  year ;  what  proportion  ought  each 
family  to  pay  ? 

7.  Divide  the  number  720  into  3  parts  in  proportion  to  3, 
4,  and  5. 

8.  Divide  $650  among  4  persons  so  that  their  shares  shall  be 
to  each  other  in  the  proportion  of  J,  \,  |,  and  y\. 

Analysis. — Since  one  part  is  \  or  VV  share,  another  \  or  -^^  share, 
another  f  or  ^^,  and  the  other  /^  of  a  share,  the  whole  is  \%  share,  and 
1  share  equals  $650 -^||  =  $300.  Hence,  \  share  is  $150,  \  share  is  $100, 
I,  $225,  yV,  $175,  Ans. 

Peoof.— $150  +  $100  +  $235  +  $175  =  $650. 

9.  Divide  945  into  3  parts  which  shall  be  to  each  other  in 
the  proportion  of  J,  -J,  -f^. 

10.  What  number  added  to  5  times  itself  will  make  576  ? 

Analysis. — A  number  added  to  5  times  itself  will  make  6  times  that 
number.  Since  576  is  the  product  of  two  factors,  one  of  which  is  6,  the 
other  factor  must  be  576-f-6  =  96,  Ans 

11.  What  number  added  to  \  of  itself  will  make  369  ? 

12.  What  number  added  to  4|-  times  itself  will  make  60-^  ? 

13.  A  man  being  asked  how  far  he  had  walked,  replied  that 
the  number  of  kilometers  he  had  traveled  was  364,  and  he  had 
ridden  twice  as  far  as  he  had  walked ;  how  many  kilometers 
had  he  walked  ? 

14.  A  lad  bought  apples,  pears,  and  peaches,  in  all  280 ;  the 
number  of  his  apples  was  twice  that  of  his  pears,  and  the  num- 
ber of  his  pears  was  twice  that  of  his  peaches ;  how  many  of 
each  did  he  buy  ? 

15.  Divide  192  into  three  such  parts  that  the  first  shall  be 
twice  the  second^  and  the  third  three  times  the  second. 


270  General  Analysis. 

652.  16.  If  4  men  can  mow  48  acres  of  grass  in  5  days,  how 
long  will  it  take  9  men  to  mow  60  acres  ? 

Analysis. — Since  4  men  can  mow  48  acres  in  5  d.,  1  man  can  mow  \  of 
48  A.,  or  12  A.  Now  if  1  man  can  mow  12  A.  in  5  d.,  in  1  d.  lie  can  mow 
\  of  12,  or  2|  A.  Again,  since  to  mow  2|  A.  requires  1  man  1  d.,  60  A. 
will  require  liira  as  many  days  as  2|  are  contained  times  in  60  =  25  d. ; 
and  since  it  takes  1  man  25  d.,  it  will  take  9  men  i  of  25  =  2|  d,,  Ans. 

17.  If  14  men  can  build  84  rods  of  wall  in  3  days,  how  long 
will  it  take  20  men  to  build  300  rods  ? 

18.  If  1000  hektoliters  of  provisions  will  support  a  garrison  of 
75  men  for  3  months,  how  long  will  3000  hektoliters  support  a 
garrison  of  300  ? 

19.  If  7  men  can  reap  42  acres  in  6  days,  how  many  men  will 
it  take  to  reap  100  acres  in  5  days  ? 

20.  If  a  man  travels  320  miles  in  10  days,  traveling  8  hours 
per  day,  how  far  can  he  go  in  15  days,  traveling  12  hours  per 
day  ? 

21.  If  24  horses  eat  126  bushels  of  oats  in  36  days,  how  many 
bushels  will  32  horses  eat  in  48  days  ? 

653.  22.  A  farmer  wishes  to  mix  a  quantity  of  corn  worth 
75  cts.  a  bushel,  with  oats  worth  37J-  cts.  a  bushel,  so  that  the 
mixture  may  be  worth  50  cts.  a  bushel ;  what  part  of  each 
must  he  take  ? 

Analysis. — Since  the  mixture  is  worth  50  cts.  a  bushel,  on  every 
bushel  of  corn  he  puts  in,  the  loss  is  25  cts.,  and  on  every  bushel  of  oats, 
the  gain  is  12|  cts.  Since  it  requires  1  bu.  oats  to  gain  12^  cts.,  to  gain 
25  cts.  will  require  as  many  bu.  of  oats  as  124  cts.  are  contained  times  in 
25  cts.,  and  25-4-12^  =  2.     Hence,  he  must  take  2  bu.  oats  to  1  bu.  corn. 

Proof.— A  mixture  of  3  bushels  is  worth  37|  cts. +  37i  cts. +  75  cts.  — 
$1.50  ;  hence,  1  bu,  mixture  is  worth  50  cents. 

Note. — The  principle  by  which  this  and  similar  examples  are  solved, 
is  that  the  excess  of  one  article  alove  the  mean  price  of  the  mixture,  coun- 
terbalances the  deficiency  of  another  article  which  is  below  it. 

23.  A  tea  merchant  has  two  kinds  of  tea  worth  40  cts.  and 
90  cts.  a  pound,  and  wishes  to  make  a  mixture  worth  60  cts.  a 
pound  ;  what  part  of  each  must  he  take  ? 

24.  How  much  ginger  at  24  cts.  and  30  cts.  a  pound,  will 
form  a  mixture  worth  25  cts,  a  pound  ? 


Definitions. 

654.  Ratio  is  the  relation  of  one  number  to  another.     It  is 
found  by  dividing  one  by  the  other. 

Thus,  tlie  ratio  of  6  to  3  is  6  h-  3,  and  is  equal  to  3. 

655.  The  Terms  of  a  Ratio  are  the  numbers  compared. 

656.  The  Antecedent  of  a  ratio  is  the  Jirst  term. 

657.  The  Consequent  is  the  second  term. 

658.  The  two  terms  together  are  called  a  Couplet. 

Thus,  in  the  ratio  9:3,  9  is  the  antecedent,  3  the  consequent,  and  9  and  3 
together  form  a  couplet. 

659.  Ratio  is  commonly  denoted  by  a  colon  ( : ),  whicli  is  a 
contraction  of  the  sign  of  division. 

Thus,  the  ratio  of  6  to  3  is  written  "  6  :  3,"  and  is  equivalent  to  6 -=-3. 

660.  Ratio  is  also  denoted  by  writing  the  consequent  under 
the  antecendent  in  the  form  of  a  fraction. 

Thus,  the  ratio  of  8  to  4  is  written  | ,  and  is  equivalent  to  8  :  4. 

Oral     Exercises. 

661.  1.  What  is  the  ratio  of  48  :  6  ?     Of  63  :  7  ?     Of  72  :  8  ? 

2.  What  is  the  ratio  of  21 :  42  ?     Of  15  :  45  ?     Of  25  to  100  ? 

3.  What  is  the  ratio  of  8  lb.  to  40  lb.  ?     Of  54  yd.  to  6  yd.  ? 

4.  What  is  the  ratio  of  150  :  llO  ?     Of  $25  :  $100  ? 

5.  Find  the  ratio  of  6  ft. :  3  hr. 

Ans.  The  ratio  cannot  be  found,  because  one  of  these  num- 
bers is  neither  equal  to  nor  a  part  of  the  other.     Hence,  the 

Principle. 

662.  Only  like  numbers  can  he  compared  with  each  other. 


272  Ratio, 

663.  A  Simple  Ratio  is  the  ratio  of  two  numbers,  as  8  : 4. 

664.  A  Compound  Ratio  is  the  product  of  two  or  more  sim- 
ple ratios.  They  are  commonly  denoted  by  placing  the  simple 
ratios  under  each  other. 

Thus,  4:2)        ^noo-  j^- 

^  >  or,  4  X  9  :  2  X  d,  IS  a  compound  ratio. 
9  :  o  ) 

665.  A  Compound  Ratio  is  reduced  to  a  simple  one  by  mak- 
ing the  product  of  the  antecedents  a  new  antecedent^  and  the 
product  of  the  consequents  a  new  consequent. 

666.  A  Reciprocal  of  a  Ratio  is  a  simple  ratio  inverted,  and 
is  the  same  as  the  ratio  of  the  reciprocals  of  the  two  numbers 
compared. 

Thus,  the  reciprocal  of  8  to  4  is  |  to  |-  =  4  :  8,  or  |. 

Note. — 1.  Reciprocal  Ratio  is  sometimes  called  Inverse  Ratio. 

2.  The  reciprocal  of  a  ratio,  when  a  fraction  is  used,  is  expressed  by- 
inverting  the  terms  of  the  fraction  which  expresses  the  simple  ratio. 
Wlien  the  colon  is  used,  the  order  of  the  terms  is  inverted. 

667.  The  ratio  between  two  fractions  which  have  a  common 
denominator,  is  the  same  as  the  ratio  of  their  numerators. 

Thus,  the  ratio  of  f  :  f  is  the  same  as  6  :  3. 

Note. — When  the  fractions  have  different  denominators,  reduce  them 
to  a  common  denominator ;  then  compare  their  numerators.  Compound 
numbers  must  be  reduced  to  the  swme  denomination. 


Written     Exercises. 

668.  Find  the  following  ratios  in  the  lowest  terms  : 
1.     95  to  25.  5.     65  to  180.  9.     f|  to  f^. 


2. 

110  to  48. 

6. 

84  to  132. 

10. 

«  to  If. 

3. 

135  to  51. 

7. 

108  to  256. 

11. 

M  to  ^. 

4. 

186  to  84. 

8. 

220  to  500. 

12. 

^  to  4^. 

13.  96  gal.  to  24  qt.  17.  15s.  to  4s.  6d. 

14.  75  bu.  to  160  pk.  18.  10  ounces  to  95  pounds. 

15.  140  rd.  to  20  ft.  19.  8  yards  to  9  inches. 

16.  175s.  to  130d.  20.  3  pnits  to  4  gallons. 


Ratio.  273 

669.  Since  the  antecedent  corresponds  to  the  numerator,  and 
the  consequent  to  the  denominator,  changes  on  the  terms  of  a 
ratio,  have  the  same  effect  upon  its  value  as  like  changes  have 
upon  the  terms  of  a  fraction.     Hence,  the 

Principles. 

i°.  Mtdtipluinq  the  antecedent,  or)  ,,  ,,.  ,. 

T^.  .t      jX  j_  r  Multiplies  the  ratio. 

Dividing  the  consequent.  )  ^ 

2°.  Dividinq  the  antecedent,  or       \  t^.  . .     ,-, 

,^  ,,.  ,   .      ^,  ,       \  Divides  the  ratio. 

MuUijnying  the  consequent,      ) 

S°.  Multiplying  or  dividing  hoth  \  Does  not  alter  the  value 
terms  iy  the  same  quantity, )      of  the  ratio. 

670.  The  ratio,  antecedent,  and  consequent  are  so  related  to 
each  other,  that  if  any  two  of  them  are  given  the  other  may 
be  found.     Hence,  the 

i  1.   The  Ratio  =  Antecedejit  -^  Consequent. 

Formulas.  I  2.   The  Consequent  =  Antecedent  -i-  Ratio. 

(  3.   The  Antecedent  =  Consequent  x  Ratio. 

21.  What  is  the  ratio,  when  the  antecedent  is  63  and  the 
consequent  9  ? 

22.  When  the  antecedent  is  25  and  the  consequent  60,  what 
is  the  ratio  ? 

23.  If  the  antecedent  is  8  and  the  ratio  14,  w^hat  is  the  con- 
sequent ? 

24.  When  the  consequent  is  16  and  the  ratio  7,  what  is  the 
antecedent  ? 

25.  When  the  antecedent  is  6J  and  the  consequent  8,  what 
is  the  ratio  ? 

26.  If  the  antecedent  is  5^  and  the  ratio  9f,  what  is  the 
consequent  ? 

27.  When  the  consequent  is  24  and  ratio  8,  what  is  the 
antecedent  ? 

28.  If  the  consequent  is  36  and  ratio  12,  what  is  the 
antecedent  ? 

29.  When  the  antecedent  is  9|  and  ratio  8|,  what  is  the 
consequent  ? 


ROPORTION. 


Oral     Exercises. 

671.  1.  What  is  the  ratio  of  12  to  60  ?     Of  9  to  36  ? 

2.  Which  of  the  above  ratios  is  the  larger  ? 

3.  How  does  the  ratio  of  30  to  6  compare  with  15  to  3  ? 

4.  How  does  the  ratio  of  5  to  25  compare  with  the  ratio  of 
12  to  60  ? 

5.  Are  all  ratios  equal  ? 

6.  How  does  the  ratio  of  12  to  4  compare  with  the  ratio  of 
10  to  5  ? 

7.  Name  two  equal  ratios.     Name  tAvo  others. 

8.  Name  two  unequal  ratios.     Name  two  others  ? 

9.  What  two  numbers  have  the  same  quotient  as  16  divided 
by  4  ?     As  28  divided  by  4  ?     As  60  divided  by  12  ? 

10.  Express  in  both  forms  the  ratio  of  two  numbers  which 
have  the  same  ratio  as  6  to  12. 

11.  How  does  the  ratio  of  7  to  14  compare  with  the  ratio  of 
5  to  20  ? 

12.  How  does  the  ratio  of  $15  to  15  compare  with  the  ratio 
of  18  ft.  to  6  ft.  ? 

Ans.  They  are  equal  to  each  other. 

672.  Proportion  is  an  equality  of  ratios. 

Thus,  the  ratio  8  :  4  =  6  :  3,  is  a  proportion.     That  is, 
Four  quantities  are  in  propoi'tion,  when  the  first  is  the  same  multiple  or 
part  of  the  second,  that  the  third  is  of  the  fourth. 

673.  The  Sign  of  Proportion  is  a  double  colon  ( : : ),  or  the 
sign  (  =  ). 

Thus,  the  proportion  above  is  expressed  8  :  4  : :  6  :  8.     Or,  8  : 4  =  6  :  8 

l^he  first  form  is  read  "  8  is  to  4  as  G  to  3." 

The  second  is  read  "  the  ratio  of  8  to  4  equals  the  ratio  of  6  to  3." 


Proportion,  275 

674.  The  Terms  of  a  proportion  are  the  numbers  compared. 

675.  The  Antecedents  of  a  proportion  are  ^(i  first  and  third 
terms. 

676.  The  Consequents  are  the  second  dmdi  fourth  terms. 

Thus,  in  the  proportion  4:8::  3:6,  the  4  and  3  are  the  antecedents, 
and  8  and  6  the  consequents. 

677.  In  every  proportion  there  must  be  at  least  four  terms  ; 
for  the  equaUty  is  between  tivo  or  more  ratios,  and  each  ratio 
has  two  terms. 

678.  A  proportion  may,  however,  be  formed  from  three 
numbers,  for  one  of  the  numbers  may  be  repeated,  so  as  to 
form  ttvo  terms ;  as,  2  :  4  : :  4  :  8. 

Note, — When  a  proportion  is  formed  of  three  numbers,  the  middle 
number  is  called  a  mean  proportional. 

679.  The  Extremes  of  a  proportion  are  the  first  and  last 
terms. 

680.  The  Means  are  the  tivo  middle  terms. 

Thus,  in  the  proportion  9  :  12  : :  18  :  24,  9  and  24  are  the  extremes,  12 
and  18  the  means. 

Eead  the  following  :     (Art.  673.) 


1. 

35  :  7  =r  GO  :  12. 

7. 

18  :  54  : 

:  21  :  63. 

2. 

42  :  14  =  75  :  25. 

8. 

23  :  92  : 

:  34  :  136. 

3. 

72  :  24  =  168  :  56. 

9. 

37  :  148 

: :  41  :  164. 

4. 

144:  1  ::  1728:12. 

10. 

3.9   •  . 
8-14  • • 

2  .  3 

"8  •  T- 

5.  20  :  143y3g  : :  2|  :  17.  11.     816.05  :  85.35  : :  827.03  :  89.01. 

6.  4^  :  54  : :  6  :  72.  12.     60  :  15  : :  80  :  20. 

681.  The  relation  of  the  four  terms  of  a  proportion  to  each 
other  is  such,  that  if  any  three  of  them  are  given,  the  other  or 
missing  term  may  be  found. 


276  Proportion. 

J)ErELOP3I  ENT      OF      I*  R  I N  C I P  L  E  S  . 

682.  1.  If  the  first  three  terms  of  a  proportion  are  %,  4^ 
and  5,  what  is  the  fourth  or  missing  term  ? 

Analysis. — Representing  the  missing  term  by  x,  then  tlie  proportion  is 

2      6 
2  : 4  : :  6  :  a",  and  the  ratio  -a=--    These  fractions  reduced  to  a  common 

2x3?      4x6 
denom.  become  - — ;  =  -r — - ;  hence  the  numerators  are  equal.     (Art.  667.) 

rt  X  t*/  rt  X  it/ 

But  2  X  .1'  is  the  product  of  the  extremes  and  4x6  the  product  of  the 
means.  CanceUing  the  factor  2,  which  is  common  to  both,  a;  =  2  x  6,  or  12, 
is  the  missing  term  required. 

683.  From  the  preceding  example  we  derive  the  following 

Principles. 

i°.  Bi  every  proportion  the  product  of  tlie  extremes  is  equal 
to  the  jjrodud  of  the  means. 

2°.  The  2^^oduct  of  the  extremes  divided  ly  either  of  the 
means,  gives  the  other  mean. 

3°.   The  product  of  the  means  divided  ly  either  extreme,  gives 
the  other  extreme. 

684.  Find  the  missing  term  in  the  following : 

2.  12  :  42  =  20  :  x.  8.  400  rd.  :  56  rd.  =  195  :  x. 

3.  9  :  153  =  150  :  x.        9.  r.  :  400  vests  =  $87.50  :  IIOOO. 

4.  175  :  $900  =  x:S5.  10.  130  lb.  :  x  =  $150  :  1850. 

5.  x:4:0  =  120  :  100.     11.  40  gal.  :  x  =  180  :  60. 

6.  24r:x  =  12  :  144.       12.  16  yd.  :  10  ft.  =  72  :  a:. 

7.  187.5  :  7i  =  ^  :  15.    13.  x  :  75  =  |  :  f. 


Simple    Proportion. 

685.  Simple  Proportion  is  an  equality  of  two  simple  ratios. 

Note.— Of  the  three  given  numbers,  two  must  always  be  of  the  same 
kind,  and  the  third  the  same  as  the  answer  required. 


Simjple  Pro])ortion,  277 

Written    Exercises. 

686.  To  solve  problems  by  Simple  Proportion  or  by  Analysis. 

1.  If  15  books  cost  $45,  what  will  be  the  cost  of  80  books  ? 

BT  ANALYSIS. 

OPERATION. 

Since  15  books  cost  $45,  1  book  costs  J^  of  $45,  ^ 

and   80  books  will  cost   80   times   as  much,  or  4^  X  oO  __  ^^^ 

$240,  Am.  ^$ 

Ans.  $240. 

BY  PEOPOKTION.  )  ^' 

As  the  answer  is  to  be  money,  we  operation. 

make   $45   the    third  term  ;   and  as  15  :  80   : :   $45  :  Ans. 

the  cost  of  80  books  will  be  more  3 

than  that  of  15  books,  we  make  80  ^^  y^  80 

the   second  term   and   15   the  first.        And  — ^ —  =  $240,  Ans, 

We  now  have  the  two  means  and 

one  extreme,  to  find  the  other  extreme.  Hence,  we  divide  the  product  of 
the  means  by  the  given  extreme,  and  the  quotient  is  the  other  extreme  or 
answer.     (Art.  683.) 

Proof.— 80  x  45  =  240  x  15.    (Art.  683,  i°.) 

687.  From  the  preceding  principles  we  have  the  following 

Rule. — I.  Malce  that  number  the  third  term,  ivhich  is 
the  same  hind  as  the  answer. 

II.  When  the  answer  is  to  he  larger  than  the  hhird 
term,  mahe  the  larger  of  the  other  two  numbevs  the 
second  term;  but  when  less,  jjlace  the  smaller  for  the 
second  term,  and  the  other  for  the  first. 

III.  Multiply  the  second  and  third  terms  together,  and 
divide  the  product  by  the  first ;  the  quotient  will  be  the 
fourth  teJin  or  answer. 

Proof. — //  the  product  of  the  first  and  fourth  terms 
equals  that  of  the  second  and  third,  the  answer  is  right. 

Notes. — 1.  The  arrangement  of  the  terms  in  the  form  of  a  proptyrtion 
is  called  "  The  Statement  of  the  question." 

2.  The  factors  common  to  the  first  and  second,  or  to  the  first  and  tMrd^ 
terms,  should  be  cancelled. 


278  Proportion. 

3.  The  fij'st  and  secoiid  terms  must  be  reduced  to  the  same  denomina- 
tion. The  third  term,  if  a  compound  number,  must  be  reduced  to  the 
lowest  deno7Mnation  it  contains. 

4.  It  is  advisable  for  the  pupil  to  solve  the  following  examples  both  by- 
Proportion  and  by  Analysis. 

2.  If  14  vests  are  worth  184,  what  are  23  vests  worth  ? 


By  Analysis. — If  14  vests  are  worth  $84,  1  vest  is  worth  1  fourteenth 

of  $84,  ot  $6,  and  23  vests  are  worth  23  times  $6,  or  $138,  Ans. 

By  PfiOPORTiON.— 14  V.  :  23  V.  : :  $84  :  Ans.  That  is,  14  is  the  same 
part  ot  23  as  $84  are  of  the  cost  of  23  vests.     Cancelling,  etc., 

6 

1^  :  23  : :  %U  :  1138,  Ans. 

3.  If  5  men  can  mow  a  meadow  in  6  days,  how  long  will  it 
take  8  men  to  mow  it  ? 

4.  If  6  acres  and  40  rods  of  land  cost  1125,  how  much  will 
25  acres  and  120  rods  cost  ? 

5.  If  15  meters  of  silk  cost  £4  10s.,  what  will  75  meters 
cost  ? 

6.  If  a  railroad  car  goes  35  mi.  in  1  hr.  45  min.,  how  far  will 
it  go  in  3  days  ? 

7.  If  84  lbs.  of  cheese  cost  $5|,  what  will  60  lbs.  cost  ? 

8.  If  f  of  a  ship  is  worth  $6000,  liow  much  is  ^  of  her 
worth  ? 

9.  If  a  ship  has  sufficient  water  to  last  a  crew  of  25  men  for 
8  months,  how  long  will  it  last  15  men  ? 

10.  If  the  interest  of  $1500  for  12  mo.  is  $90,  what  will  be 
the  interest  of  the  same  sum  for  8  mo.  ? 

11.  If  a  tree  20  ft.  high  casts  a  shadow  30  ft.  long,  how  long 
will  be  the  shadow  of  a  tree  50  ft.  high  ? 

12.  How  long  will  it  take  a  steamship  to  sail  round  the 
globe,  allowing  it  to  be  25000  miles  in  circumference,  if  she 
sails  at  the  rate  of  3000  miles  in  12  days  ? 

13.  How  many  hektars  of  land  can  a  man  buy  for  $840,  if  he 
pays  at  the  rate  of  $56  for  every  7  hektars? 


Cau8e  and  Effect  279 


Cause    and    Effect. 

688.  The  principles  of  proportion  may  also  be  explained  by 
clie  relations  of  the  terms  to  each  other,  as  causes  and  effects, 

689.  A  Cause  is  that  which  produces  something. 

An  Effect  is  something  which  is  produced. 

Thus,  men  at  work,  goods  bought  or  sold,  time,  money  lent,  etc.,  are 
causes.     Work  done,  provisions  consumed,  cost  of  goods,  etc. ,  are  effects. 

690.  In  arithmetical  operations  it  is  assumed  that  like  causes 
produce  like  effects,  and  the  ratio  between  the  effects  is  equal 
to  the  ratio  between  the  causes  which  produce  them. 

If  2  horses  as  a  cause  can  move  3  tons  as  an  effect,  6  horses 
as  a  cause  will  remove  9  tons  as  an  effect ;  that  is 

2  horses  (1st  C.) :  6  horses  (2d  C.)  : :  3  T.  (1st  E.) :  9  T.  (2d  E.) 

Written     Exercises. 

691.  1.  If  4  acres  produce  60  bushels  of  wheat,  how  nauch 
will  9  acres  produce  ? 

Analysis. — In  this  example, 
the  two  causes  are  4  acres  and  9 
acres;  the  first  effect  is  60  bu., 
the  second  effect  is  required. 

We  make  60  bu.  the  given 
effect,  the  third  term,  and  since  the  second  effect  must  be  greater  than  the 
first,  we  make  9  A.,  the  greater  cause,  the  second  term,  and  4  A.  the  first. 
Multiplying  and  dividing  as  before,  the  result  is  135  bu.,  Ans. 

2.  If  it  requires  4  acres  to  produce  60  bushels  of  wheat,  how 
many  acres  are  required  to  produce  135  bushels  ? 

Analysts. — In  this  example  operation. 

two  effects,  60  bii.  and  185  bu.,  1st.  E.         2d  E.             1st  C. 

and  one  cause,  4  A.,  are  given,  60  bu.  :  loO  DU.  :  :  4  A.  :  Ans. 

the    second    cause    is    required.  (135  X  4)  -J-  60  =  9 

Since  the  2d  effect  is  greater  than  Ans     9  acres 
the  first,  the  2d  cause  must  also 

be  greater  than  the  given  cause ;  we  therefore  make  135  bu.  the  2d  term 

and  60  bu.  the  1st  term.     The  result  is  9  acres. 


1st  C.      2d  C. 

IstE. 

4  A.  :  9  A. 

: :  60  bu.  :  Ans. 

(60  X  9)  . 

-^  4  =  135 

Ans.  135  bu. 

280  Proj[>ortion. 

692.  When  the  terms  of  a  proportion  are  considered  in  the 
relation  of  cause  and  effect^  the  operations  are  the  same  as  when 
considered  in  the  relation  of  magnitude.     (Arts.  687,  689.) 

Solve  the  following  by  either  or  both  the  preceding  methods: 

3.  Bought  41  yd.  of  flannel  for  116.40;  how  much  would 
8|  yd.  cost  ? 

4.  Bought  18  kilos  of  ginger  for  18.50;  how  much  will  10| 
kilos  cost  ? 

5.  If  a  stage  goes  84  kilometers  in  12  hours,  how  far  will  it 
go  in  15|-  hours  ? 

6.  If  26  horses  eat  72  hektoliters  of  oats  in  a  week,  how  many 
hektoliters  will  25  horses  eat  in  the  same  time  ? 

7.  If  a  railroad  car  runs  125  kilometers  in  5  hours,  how  far 
will  it  run  in  12|  hours  ? 

8.  If  9  ounces  of  silver  will  make  4  tea  spoons,  how  many 
spoons  will  25  pounds  of  silver  make  ? 

9.  If  5J  yd.  of  cloth  are  worth  |;27J,  what  are  50^  yd.  worth? 

10.  If  60  men  can  build  a  house  in  90^  days,  how  long  will 
it  take  15  men  to  build  it  ? 

11.  What  will  49 1\  yd.  velvet  cost,  if  7f  yd. 'cost  £7  ISs.  4d.  ? 

12.  At  7s.  6d.  per  ounce,  what  is  the  value  of  a  silver  pitcher 
weighing  9  oz.  13  pwt.  8  gr.  ? 

13.  If  405  yd.  linen  cost  £69  7s.  6d.,  what  cost  243  yd  ? 

14.  If  A  can  saw  a  cord  of  wood  in  6  hours,  and  B  in  10 
hours,  how  long  will  it  take  both  together  to  saw  a  cord? 

15.  A  cistern  has  3  stop-cocks,  the  first  of  which  will  emj)ty 
it  in  10  min. ;  the  second,  in  15  min. ;  and  the  third,  in  30 
min.;  how  long  will  it  take  all  of  them  together  to  empty  it? 

16.  A  man  and  a  boy  together  can  mow  an  acre  of  grass  in 
4  hours  ;  the  man  can  mow  it  alone  in  6  hours ;  how  long  will 
it  take  the  boy  to  mow  it  ? 

17.  If  the  interest  of  $675.25  is  155.625  for  1  year,  how 
much  will  be  the  interest  of  $2368.85  ? 

18.  What  must  be  the  length  of  a  board  which  is  9f  in.  wide, 
to  make  a  square  foot  ? 

19.  If  98J  yds.  carpeting  \\  yard  wide  will  cover  a  floor,  how 
many  yards  f  yd.  wide  will  it  take  to  cover  it  ? 


Compound  Proportion,  281 


Compound    Proportion. 

693.  Compound  Proportion  is  an  equality  between  a  com- 
pound ratio  and  a  simple  one.     Thus, 

8  :  4) 

h  • :  13  :  3,  is  a  compound  proportion.     That  is, 
D  :  o ' 

8x6:4x3::  12  :  3  ;  for,  8  x  6  x  3  =  4  x  3  x  12. 
It  is  read,  '^  The  ratio  of  8  into  6  is  to  4  into  3,  as  12  to  3." 

Written     Exercises. 

694.  1.  If  4  men  earn  $60  in  10  days,  liow  much  can  6  men 
earn  in  8  days  ? 

Explanation. — Since  the  answer  operation. 

is  to   be  money,  we  make   $60  the  4  m.  :  6  m. 

third  term. 

other  numbers  in  pairs,  two  of  a 
kind,  placing  them  according  as  the  answer  would  be  greater  or  less  than 
the  third  term,  if  it  depended  on  each  pair  alone.  Now,  as  6  m.  can  earn 
more  than  4  m.,  we  place  the  larger  for  the  second  term  and  the  smaller 
for  the  first.  Agahi,  as  they  will  earn  less  in  8  d.  than  in  10  d.,  we  place 
the  smaller  for  the  second  term,  and  the  large]'  for  the  first. 

Reducing  the  compound  ratio  to  a  simple  one,  we  have, 

4  X  10  :  6  X  8  : :  60  :  Ans. 
Dividing  the  prod,  of  the  means  by  the  extreme,  cancelling,  etc. 


ley,  we  make   $60  the  4  m.  :  6  m.  )  _ 

We   then    arrange   the         10  d     *  8  d     ("  •  '   '*^^'-'  *  ■^^^^' 


^^^^  =  172,  Ans. 

^  X  10 

695.  From  the  preceding  example  we  have  the  following 

Rule. — I.  Make  that  niuiiber  which  is  of  the  same 
kind  as  the  ansiuei^  the  thiixl  teriiv. 

II.  Tlien  take  the  other  ninnhers  in  pairs,  or  two  of  a 
kind,  and  arrange  theiyv  as  in  simple  proportion. 
(Art.  687.) 

III.  Maltiply  the  second  and  third  terms  together,  and 
divide  the  product  by  the  product  of  the  first  terms. 
TIxe  quotient  will  l>e  the  answer. 


282  Proportion, 

Proof. — If  the  product  of  the  first  and  fourth  terms 
equals  that  of  the  second  and  third  terms,  the  luorh  is 
right. 

Notes. — 1.  The  terms  of  eacli  couplet  in  the  compound  ratio  must  be 
reduced  to  the  same  denomination,  and  the  third  term  to  the  lowest 
denomination  contained  in  it,  as  in  Simple  Proportion, 

2.  In  Compound  Proportion,  all  the  terms  are  given  in  couplets  ot  pairs 
of  the  same  kind,  except  one.  This  is  called  the  odd  term,  or  demand,  and 
is  always  the  same  kind  as  the  answer. 

3.  Problems  in  Compound  Proportion  may  also  be  solved  by  Analysis 
and  by  Simple  Proportion.     Take  the  preceding  example. 

By  Analysis. — If  4  men  can  earn  $60  in  10  d.,  1  man  can  earn  in  the 
same  time,  \  of  $60,  which  is  $15,  and  6  men  can  earn  6  times  15  or  $90. 

Again,  if  6  men  earn  $90  in  10  d.,  in  1  d.  they  can  earn  ^l  of  $90,  which 
is  $9  ;  and  in  8  d.  they  can  earn  8  times  9,  or  $72,  Ans. 

By  Simple  Propoktion.— 4  m.  :  6  m.  : :  $60  :  x,  or  $90. 
Again,  10  d.  :  8  d.    : :  $90  :  Ans.,  or  $72. 

2.  If  8  men  can  clear  30  acres  of  land  in  C3  days,  working 
10  hours  a  day,  how  many  acres  can  10  men  clear  in  72  days, 
working  12  hours  a  day  ? 


STATEMENT. 


8  m.    :10  m.   |     ^^^^^  ,  ^3 

.2  br.  ) 

OPEBATION. 


63  d.    :  72  d.     V  : :  30  :  A7is.  ^0 

10  hr.  :  12  br.  ) 


10  X  72  X  12  X  30        .,,  ,       , 
8  X  63  X  10        =  '^^  ^'^  ^'^^• 


10 

12 
30 

360  =  51f  A.,  Ans. 


Note. — When  the  vertical  form  of  cancellation  is  used,  the  antecedents 
must  be  placed  on  the  left  of  the  line,  and  the  consequents  with  the  odd 
term  on  the  right. 

3.  If  a  man  can  walk  192  miles  in  4  days,  traveling  12  hours 
a  day,  how  far  can  he  go  in  24  days,  traveling  8  hours  a  day  ? 

4.  If  8  men  can  make  9  rods  of  wall  in  12  days,  how  many 
men  will  it  require  to  make  36  rods  in  4  days  ? 


Comjpoiind  Proportion,  283 

5.  If  5  men  make  240  pair  of  shoes  in  24  days,  how  many 
men  will  it  require  to  make  300  pair  in  15  days  ? 

6.  If  60  lbs.  of  meat  will  supply  8  men  15  days,  how  long 
will  72  lbs.  last  24  men  ? 

7.  If  12  men  can  reap  80  acres  of  wheat  in  6  days,  how  long 
will  it  take  25  men  to  reap  200  acres  ? 

8.  If  18  horses  eat  128  bushels  of  oats  in  32  days,  how  many 
bushels  will  12  horses  eat  in  64  days  ? 

9.  If  8  men  can  build  a  wall  20  ft,  long,  6  ft.  high,  and  4  ft. 
thick,  in  12  days,  how  long  will  it  take  24  men  to  build  one 
200  ft.  long,  8  ft.  high,  and  6  ft.  thick  ? 

10.  If  8  men  reap  36  acres  in  9  days,  working  9  hours  per 
day,  how  many  men  wdll  it  take  to  reap  48  acres  in  12  days, 
working  12  hours  per  day? 

11.  If  1100  gain  $6  in  12  months,  how  long  will  it  take  $400 
to  gain  118.     Ans.  9  mo. 

12.  If  $200  gain  $12  in  12  mo.,  what  will  1400  gain  in  9  mo.? 

13.  If  6  men  can  dig  a  drain  20  rods  long,  6  feet  deejo,  and 
4  feet  wide,  in  16  days,  working  9  hours  each  day,  how  many 
days  will  it  take  24  men  to  dig  a  drain  200  rods  long,  8  feet 
deep,  and  6  feet  wide,  working  8  hours  per  day  ? 

14.  If  3  lbs.  of  yarn  will  make  10  yards  of  cloth  \\  yard 
wide,  how  many  pounds  will  be  required  to  make  a  piece  100 
yards  long,  and  1^  yd.  wide  ? 

15.  A  general  wished  to  remove  80000  lbs.  of  provision  from 
a  fortress  in  9  days,  and  it  was  found  that  in  6  days  18  men 
had  carried  away  but  15  tons;  how  many  men  would  be  re- 
quired to  carry  the  remainder  in  3  days  ? 

16.  If  a  man  travels  130  miles  in  3  days,  when  the  days  are 
14  hours  long,  how  long  will  it  take  him  to  travel  390  miles 
when  the  days  are  7  hours  long? 

17.  If  the  price  of  10  oz.  of  bread  is  5d.,  when  corn  is  4s.  2d. 
per  bushel,  what  must  be  paid  for  3  lbs.  10  oz.  when  corn  is 
5s.  5d.  per  bushel  ? 

18.  If  6  journeymen  make  132  pair  of  boots  in  4^  weeks, 
working  5^  days  a  week,  and  12f  hours  j^er  day,  how  many  pair 
will  18  men  make  in  13|^  weeks,  working  ^\  days  per  week, 
and  11  hours  per  day  ? 


284  Projportion, 


Partitive    Proportion. 

696.  Partitive  Proportion  is  dividing  a  number  into  two  or 
more  parts  having  a  given  ratio  to  each  other. 

Oral     Exercises. 

697.  1.  Cliarles  and  Robert  divided  28  pears  between  them 
in  the  ratio  of  3  to  4 ;  how  many  had  each  ? 

Analysis. — Since  Cliarles  had  3  parts  as  often  as  Robert  had  4,  both 
had  3  +  4,  or  7  equal  parts.  Hence,  Charles  had  f  and  Robert  |  of  28. 
Now  f  of  28  are  12,  and  |  are  16.  Therefore,  Charles  had  12,  and  Robert 
16  pears. 

Proof. — 12  pears  +16  pears  =  28  pears, 

2.  Divide  35  into  2  such  parts  that  one  shall  be  to  the  other 
as  3  to  2. 

3.  Divide  42  cts.  into  two  such  parts  that  one  shall  be  to  the 
other  as  2  to  4. 

4.  A  farmer  had  56  acres  of  which  he  made  2  pastures  in 
the  ratio  of  3  to  5  ;  how  many  acres  were  in  each  ? 

5.  A  man  bought  a  cow  and  a  calf  for  150 ;  the  cow  was 
worth  4  times  as  much  as  the  calf ;  what  was  the  value  of 
each  ? 

6.  Divide  |72  into  two  such  parts  that  one  shall  be  to  the 
other  as  4  to  8. 

Written     Exercises. 

698.  To  divide  a  number  into  two  or  more  parts  which  shall  have 

a  given  ratio  to  each  other. 

1.  A  and  B  divided  $145  in  the  ratio  of  2  to  3 ;  how  much 
had  each  ? 

Solution. — The  sum   of  the  pro-  operation. 

portional  parts  is  to  each  separate  part  5:2  :  :   $145  :  A's  S. 

as  the  number  to  be  divided  is  to  each  5*3   :  •   1^145  :  B's  S. 

man's  share      That  is,  5  (2  +  3)  is  to  3  ^^^^        %)^r^  =  *58,'  A's  s! 

as  $145  to  A's  share.     Affam,  5  is  to  ^            '^    /   •             *      ' 

3  as  $145  to  B's  share.     Hence,  the  (^1^^"^  x  3)  -f-5  =  $87,  B's  S. 


Partitive  Pro])ortion,  285 

EuLE. — I.  Mahe  the  niunher  to  he  divided  the  third 
term;  each  proportional  part  successively  the  second 
term  ;  and  their  sum  the  first. 

11.  T]ie  product  of  the  second  and  third  terms  of  each 
proportion,  divided  by  the  first,  will  he  the  corresponding 
part  required. 

2.  Divide  312  into  three  parts  which  shall  be  to  each  other 
as  3,  4,  and  6. 

3.  A  man  having  198  sheep,  wished  to  divide  them  into 
three  flocks  which  should  be  to  each  other  as  1,  3,  and  5 ;  how 
many  will  each  flock  contain  ? 

4.  A  farmer  raised  500  bushels  of  grain,  composed  of  oats, 
wheat,  and  corn  in  the  proportion  of  3,  4,  and  5|-;  how  many 
bushels  were  there  of  each  kind  ? 

5.  A  man  paid  15.28  for  pears,  oranges,  and  melons,  the 
prices  of  which  were  as  2,  4,  and  6  ;  how  much  did  he  pay  for 
each  kind  ? 

6.  A  father  divided  13479  among  his  four  sons  in  proportion 
to  their  ages,  which  were  as  4,  6,  8,  and  10 ;  how  many  dollars 
did  each  receive  ? 

Q  U  ESTI  ON  S. 

654.  What  is  ratio  ?  How  found  ?  655.  What  are  the  terms  of  a  ratio  ? 
656.  The  first  terra  called?  657.  The  second?  658.  The  two  terms? 
659.    Ratio    denoted?  662.  What  numbers  can  be  compared?    663. 

A  simple  ratio  ?  664.  Compound  ?  666.  Reciprocal  ?  669.  Name  the  prin- 
ciples of  ratio  ? 

673.  What  is  proportion  ?  673.  The  sigfn  ?  674.  The  terms  ?  677. 
How  many  terms  must  there  be  in  a  proportion?  678.  What  is  a  mean 
proportional  ?  679.  The  extremes  ?  680.  The  means  ?  683.  Principles  of 
proportion  ? 

685.  What  is  a  simple  proportion  ?  687.  Which  number  is  the  third 
term  ?    How  arrange  the  other  terms  ?    How  find  the  fourth  term  ? 

688.  How  else  may  proportion  be  explained  ?  689.  What  is  a  cause  ? 
An  effect  ? 

693.  What  is  compound  proportion  ?  695.  Which  number  is  made  the 
third  term?  How  arrano-e  the  remaining  numbers?  How  find  the  re- 
quired term  ?  69o.  What  is  partitive  proportion  ?  698.  How  arrange  the 
terms  ?    How  find  the  answer  ? 


ARTNERSHiP 


699.  Partnership  is  the  association  of  two  or  more  persons 
for  the  transaction  of  business. 

700.  The  association  is  called  a  Firm,  Company,  or  House. 

701.  The  persons  associated  are  called  Partners. 

702.  Tlie  Capital  is  the  money  or  property  furnished  by  the 
Partners. 

703.  The  Assets  of  a  firm  are  the  various  kinds  of  property 
in  its  possession. 

704.  The  Liabilities  are  its  debts. 

705.  The  Net  Capital  is  the  excess  of  its  property  above  its 
liabilities. 

Written     Exercises. 

706.  To  find   each    Partner's   Share  of  the  Profit  or  Loss,  when 

their  capital  is  employed  for  the  same  time. 

1.  A  and  B  formed"  a  partnership  ;  A  put  in  1400  and  B 
$300 ;  they  make  1364.  What  was  each  man's  share  of  the 
profit  ? 

By  Analysis.— $400  +  $300  =  $700,  the  whole  capital.     Hence, 
A  had  |g-a,  or  f  of  the  gain ;  and  |  of  $364  =  $208,  A's. 
B  had  f  ga,  or  f    "  "        "     f  of  $364  =  $156,  B's.     Hence, 

EuLE. — I.  Take  such  a  part  of  the  gain  or  loss,  as  each 
partner's  stock  is  of  the  ivJiole  capiial. 

By  Percentage.— The  gain  $364  is  fUJ  =  ^^^,  or  53%  of  the  whole 
capital.     Therefore, 

X  .53  =  $308,  A's  share  ;  and  $300  x  .53  =  $156,  B's  share.    Hence, 


II.  Find  the  %  v;hich  the  profit  or  loss  is  of  the  whole 
capital,  and  multiply  each  man's  capital  hy  it.    (Art.  464.) 


Partnership.  287 

Note. — The  per  cent  method  is  preferable,  when  the  partners  or  share- 
holders are  numerous. 

2.  A  and  B  formed  a  partnership ;  A  put  in  $648  and  B 
$1080,  agreeing  to  divide  the  profit  in  proportion  to  their 
capital ;  what  was  each  one's  share  of  the  gain  ? 

3.  A,  B,  and  C  are  partners ;  A  furnishes  $600,  B  $800,  and 
0  $1000  ;  they  lose  $480  ;  what  is  each  man's  share  of  the  loss  ? 

4.  A,  B,  0,  and  D  make  up  a  purse  for  a  stock  speculation  ; 
A  puts  in  $300,  B  $400,  C  $600,  and  D  $800  ;  they  make 
$2400  ;  what  is  each  man's  share  ? 

707.  To  find  each  partner's  share  of  the   Profit  or  Loss,  when 

their  capita!  is  employed  for  unequal  times. 

5.  A  and  B  enter  into  partnership;  A  furnishes  $400  for 
8  months,  and  B  $600  for  4  months  ;  they  gain  $280  ;  what  is 
each  one's  share  of  the  profit  ? 

Analysis. — In  this  case  the  profit  of  each  partner  depends  on  two  ele- 
ments, viz.  •.  the  amount  of  his  capital  and  the  time  it  is  employed. 

But  the  use  or  interest  of  $400  for  8  months  equals  that  of  8  times  $400, 
or  $3200,  for  1  mo. ;  and  of  $600  for  4  mo.  equals  4  times  that  of  $600,  or 
$2400,  for  1  month. 

The  respective  capitals,  then,  are  equivalent  to  $2400  and  $3200,  each 
employed  for  1  mo.  Since  A  furnished  $3200  and  B  $2400,  the  whole 
•apital  =  $5600.     Now  $280h-$5600  =  .05  or  5;^.     Therefore, 

$3200  X  .05  =  $160,  A's  share. 

$2400  X  .05  =:  $120,  B's  share.     Hence,  the 

Rule. — Multiply  each  -partner's  capital  hy  the  time  it 
is  employed.  Consider  these  products  as  their  respective 
capitals,  and  proceed  as  in  the  last  article. 

708.  The  Rule  for  Partnership  is  also  applicable  to  problems 
in  Bankruptcy,  the  General  Average  of  losses  at  sea,  and  other 
distributions;  the  sum  of  the  debts  or  property  in  question 
corresponds  to  the  tvhole  amount  of  capital,  etc. 

6.  A  and  B  formed  a  partnershi]) ;  A  put  in  $300  for  2 
months,  and  B  $200  for  6  months ;  they  gained  $150  ;  what 
was  each  man's  just  share  of  the  gain  ? 


288  Partnership. 

7.  A,  B^  and  C  enter  into  partnership ;  A  puts  in  1500  for 
4  mo.,  B  $400  for  6  mo.,  and  C  $800  for  3  mo.;  they  gain 
$340  ;  what  is  each  man's  share  of  the  gain  ? 

8.  A  and  B  hire  a  pasture,  together  for  $60 ;  A  put  in  120 
sheep  foip  6  months,  and  B  put  in  180  sheep  for  4  months; 
what  should  each  pay  ? 

9.  The  firm  A,  B,  and  C  lost  $246  ;  A  had  put  in  $85  for 
8  mo.,  B  $250  for  6  mo.,  and  0  $500  for  4  mo.;  what  is  each 
man's  share  of  the  loss  ? 

10.  A  man  failing  in  business  owed  A  $1200,  B  $1800,  and 
C  $2400  ;  his  assets  were  $2700  ;  how  much  did  each  receiye  ? 

Ans.  A  received  $600,  B  $900,  and  C  $1200. 

Pnoor.— $600 +  $900 +  $1200  =  $2700,  the  assets. 

11.  A  bankrupt  owes  A  $1200,  B  $2300,  0  $3400,  and  D 
$4500 ;  his  whole  effects  are  worth  $5600 ;  how  much  will  each 
creditor  receive  ? 

12.  A  railroad  company  went  into  bankruptcy  whose  liabilities 
were  $36300,  and  assets  $12100  ;  how  much  did  the  company 
pay  on  a  dollar,  and  how  much  did  a  creditor  receive  who  ]iad 
a  claim  of  $15270  ? 

13.  A,  B,  and  C  freighted  a  vessel  with  flour  from  New  York 
to  New  Orleans ;  A  had  on  board  1200  barrels,  B  800,  and  0  400. 
On  her  passage  400  barrels  were  thrown  overboard  in  a  gale  ; 
what  was  the  average  loss  ? 

14.  A  Liverpool  packet  being  in  distress,  the  master  threw 
goods  overboard  to  the  amount  of  $10000.  The  whole  cargo 
was  valued  at  $72000,  and  the  ship  at  $28000 ;  wliat  per  cent 
loss  was  the  general  average  ;  and  how  much  was  A's  loss,  whc 
had  goods  aboard  to  the  amount  of  $15000  ? 

Q  U  ESTi  ONS. 

699.  What  is  partnership  ?  700.  What  is  the  association  called  ?  702. 
What  is  the  capital  ?     703.  Assets?     704.  Liabilities?     705.  Net  capital  ? 

706.  How  find  each  partner's  share  of  profit  or  loss,  when  their  capital 
is  employed  for  the  same  time  ?    707.  When  for  unequal  times  ? 


Jl\        -  ^ *-<—m. 

NVOLUTION. 

fe  ■:  ■  ^^^ W ^^ 

Oral     Exercises. 

709.  1.  What  is  the  product  of  3  multiplied  by  itself  ? 

2.  What  is  the  product  of  3  taken  3  times  as  a  factor  ? 

3.  What  is  the  product  of  4  taken  3  times  as  a  factor  ? 

4.  What  is  the  product  of  5  taken  twice  as  a  factor  ? 

5.  What  is  the  product  of  3  taken  4  times  as  a  factor  ? 

6.  What  is  the  product  of  f  multiplied  by  itself  ? 

7.  What  is  the  product  of  |  taken  twice  as  a  factor  ? 

8.  What  is  the  product  of  .4  taken  twice  as  a  factor  ?  Of  .3 
taken  three  times  ?     Of  .04  twice  ? 

Definitions. 

710.  Involution  is  finding  a  power  of  a  number 

711.  A  Power  of  a  number  is  the  product  of  two  or  more 
equal  factors. 

Thus,  2x2x2  =  8,  and  3x3  =  9;  8  and  9  are  powers  of  2  and  3. 

712.  Powers  are  7iamed  according  to  the  Clumber  of  times  the 
factor  is  taken  to  produce  the  given  power. 

713.  The  First  Power  is  the  number  itself. 

714.  The  Second  Power  is  the  product  of  two  equal  factors, 
and  is  called  a  Square. 

715.  The  Third  Power  is  the  product  of  three  equal  factors, 
and  is  called  a  Cube. 

Note. — The  second  power  is  called  a  square  because  the  area  of  a  square 
is  found  by  multiplying  one  of  its  sides  by  itself.  The  thii'd  power  is 
called  a  cube  because  the  contents  of  a  cube  are  found  by  taking  one  of  its 
sides  three  times  as  a  factor.     (Art.  429.) 


390  Involution, 

715.  An  Exponent  is  a  small  figure  placed  above  a  number 
on  the  right  to  denote  the  power. 

It  shows  that  the  number  above  which  it  is   placed  is  to 
be  raised  to  the  power  indicated  by  this  figure.     Thus, 

2'  =  2,  the  first  power,  or  number  itself. 
2^  =  2  X  2,  the  second  power,  or  square. 
2^  =  2  X  2  X  2,  the  third  power,  or  cube. 
2^  —  2x2x2x2,  ihe  fourth  power,  etc. 

Notes. — 1.  The  term  exponent  is  from  the  Latin  exponere,  to  represent. 
2.  The  exponent  of  the  first  power  being  1,  is  commonly  omitted. 

717.  The  expression  2^  is  read,    *^  2  raised  to  the  fourth 
power,  or  the  fourth  power  of  2." 

9.  Eead  the  following:  9^  ■1%\  25^,  245^,  SSl^o,  465i^  lOOO^^. 

10.  Read  G^x^S  25^x48^  1408— 75^,  256io^97^ 

11.  Express  the  4th  power  of  85.       13.  The  7th  power  of  340. 

12.  Express  the  5th  power  of  348.     14.   The  8th  power  of  561. 

718.  To  find  any  required  Power  of  a  Number. 

1.  What  is  the  4th  power  of  8  ? 
Solution.— 8^  =  8x8x8x8  =  4096,  Ans. 

Rule. — Tahe  the  number  as  many  tunes  as  a  factor  as 
there  are  units  in  the  exponent  of  the  required  power. 

Notes. — 1.  A  common  fraction  is  raised  to  a  power  by  involving  each 
term.     Thus,  {If  =  j%. 

2.  A  mixed  number  should  be  reduced  to  an  improper  fraction,  or  the 
fractional  part  to  a  decimal ;  then  proceed  as  above. 

Thus,  (2i)'2  =  (1)2  =  2^5  ;  or  2i  =  2.5  and  (2.5)-^  =  6.25. 

3.  All  powers  of  1  are  1  ;  for  1x1x1,  etc.  =  1. 

Raise  the  following  numbers  to  the  powers  indicated: 

2.     63.  5.     55.  8.     4.033.  11.     (1)4. 

3. 


36. 

6. 

74. 

9. 

2.00033. 

12. 

{if- 

2322. 

7. 

353. 

10. 

300.053. 

13. 

(3J)^. 

Formation  of  Squares, 


291 


Formation    of    Squares. 

719.  To  find  the  Square  of   a   Number    in  the  Terms  of   its 

Parts. 

1.  Find  the  square  of  5  in  the  terms  of  the  parts  3  and  2. 

Illustration. — Let  the  shaded  part  of  the 
diagram  represent  the  square  of  3  ;  its  contents 
are  equal  to  3  x  3,  or  9  sq.  ft. 

1st.  To  preserve  the  form  of  the  square,  equal 
additions  must  be  made  to  two  adjacent  sides ; 
for,  if  made  on  one  side,  or  on  opposite  sides, 
the  figure  will  no  longer  be  a  square. 

2d.  Since  5  is  2  more  than  3,  it  follows  that 
two  rows  of  3  squares  each,  must  be  added  at 
the  top,  and  2  rows  on  one  of  the  adjacent  sides, 
to  make  its  length  and  Irreadth  each  equal  to  5.     Now  2x3  plus  2x8 
are  12  squares,  or  twice  the  product  of  the  two  parts  2  and  3. 

But  the  diagram  wants  2  times  2  small  squares,  as  represented  by  the 
dotted  lines,  to  fill  the  upper  corner  on  the  right,  and  2  times  2  or  4  is  the 
square  of  the  second  part.  We  now  have  9  (the  sq.  of  the  1st  part),  12 
(twice  the  prod,  of  the  two  parts  8  and  2),  and  4  (the  square  of  the  2d  part.) 
But  9  +  12  +  4—25,  the  square  required. 

2.  Find  the  square  of  7  in  the  terms  of  5  and  2. 

Ans,  25  +  20  +  4.  Proof.— 7  x  7  =  49. 

3.  Find  the  square  of  25  in  the  terms  of  its  tens  and  units. 


Analysis. — The  product  of  2  tens  or 
20  by  20  is  400  (the  square  of  the  tens) ; 
20  x  5  plus  20  X  5  is  200  (twice  the  prod, 
of  the  tens  by  the  units) ;  and  5  by  5  is 
25  (the  square  of  the  units).  Now  400  + 
200  +  25  zr  625,  or  25^.     Hence,  the 


25  =     20  +  5 

25  20  +  5 

125        400  +  100 

50  +  100  +  25 

625  =  400  +  200  +  25 


Rule. — TJie  square  of  any  niunher  consisting  of  tens 
and  units  is  equal  to  the  square  of  the  tens,  plus  twice  the 
product  of  the  tens  hy  the  units,  plus  the  square  of  the 
units. 


4.  What  is  the  square  of  34  in  the  terms  of  its  tens  and  units? 


i  VOLUTION. 


Q> 


T^^ 


Oral     Exercises. 

720.  1.   What  are  the  two  equal  factors  of  9  ?     16  ?    25  ? 

2.  Name  the  two  equal  factors  of  36  ?    49  ?     64  ? 

3.  What  are  the  three  equal  factors  of  8  ?     27  ?     125  ? 

4.  Name  the  four  equal  factors  of  16  ?     Of  81  ? 

5.  Of  what  is  49  the  square  ? 

6.  Of  what  is  27  the  third  power? 

7.  Of  what  is  125  the  cube? 

Definitions. 

721.  Evolution  is  finding  a  root  of  a  number. 

722.  A  Root  is  one  of  the  equal  factors  of  a  number. 

Moots  are  named  according  to  the  number  of  equal  factors  they  contain, 

723.  The  Square  Root  is  one  of  the  tivo  equal  factoids  of  a 
number. 

Thus,  5  X  5  =  25;  therefore,  5  is  the  square  root  of  25. 

724.  The  Cube  Root  is  one  of  the  three  equal  factors  of  a 
number. 

Thus,  B  X  3  X  3  ~  37  ;  therefore,  3  is  the  cube  root  of  27,  etc. 

725.  The  character  (y^)  is  called  the  Radical  Sign.  It  is  a 
corruption  of  the  letter  R,  the  initial  of  the  Latin  radix^  a 
root. 

726.  Roots  are  denoted  in  tivo  tmys : 

1st.   By  prefixing  to  the  number  the  Radical  Sign,  witli  a 
figure  placed  over  it  called  tlie  Index  of  the  root ;  as  ^4,  <^8. 
2d.  By  ^fractional  exponent  placed  above  the  number  on  the 

right.     Thus,  9^,  27^  denote  the  square  root  of  9,  and  the  cube 
root  of  27. 


Sqiiare  Root  293 

Notes. — 1.  The  figure  over  the  radical  sign  and  the  denominator  of  the 
exponent,  denote  the  name  of  the  root, 

2.  In  expressing  the  square  root,  it  is  customary  to  use  simply  the  radical 
sign  (/y/),  the  2  being  understood.  Thus,  the  expression  /^%h  —  5,  is 
read,  "  the  square  root  of  25  =  5. " 

727.  A  Perfect  Power  is  a  number  whose  exact  root  can  be 
found;  as,  9,  16,  25,  etc. 

728.  An  Imperfect  Power  is  a  number  whose  exact  root  can 
not  be  found.     This  root  is  called  a  Surd. 

Thus,  5  is  an  imperfect  power,  and  its  square  root  2.23  +  is  a  surd. 
Note. — All  roots  as  well  aspowei's  of  1,  are  1. 
Eead  the  following  expressions  : 

8.  ^/40.        10.   119i        12.   1.5^         14.    ^256.        16.    ^^ff. 

9.  ^15.        11.  243i        13.   ^29.        15.   i^45.7.       17.   ^^|. 

18.  Express  the  cube  root  of  64  both  ways ;  the  4th  root  of 
25  ;  the  7th  root  of  81 ;  the  10th  root  of  100. 


Square    Root. 

729.  Extracting  the  Square  Root  is  finding  one  of  two  equal 
factors  of  a  number. 

730.  To  find  how  many  figures  the  Square  of  a  Number  contains. 

Illustration. — 1.  Take  1  and  9,  the  least  and  greatest  integer  that  can 
be  expressed  by  one  figure  ;  also  10  and  99,  the  least  and  greatest  that  can 
be  expressed  by  two  integral  figures,  etc.     Squaring  these  numbers, 

12  z=  1  ;     102  _  100  ;     1002  —  10000. 
92  =  81 ;    992  =  9801  ;    999^  =  998001. 

2.  Take  .1  and  .9,  the  least  and  greatest  decimals  that  can  be  expressed 
by  one  figure  ;  also  .01  and  .99,  the  least  and  greatest  that  can  be  expressed 
by  two  decimal  figures,  etc.     Squaring  these, 

.12  =  .01 ;    .012  ^  .0001  ;    OOP  .-=:  .000001. 
.92  =  .83  ;    .992  =  .9801  ;    .9992  =  .998001,  etc. 


294 


Evolution. 


731.  From  these  illustrations  we  discover  the  following 


Principles. 

i°.  Tlie  square  of  a  number  contains  twice  as  many  figures 
as  the  root,  or  twice  as  many  less  one. 

2°.  If  any  number  is  separated  into  periods  of  tiuo  figures 
each  beginniyig  with  units  'place,  the  number  of  figures  in  the 
square  root  loill  be  equal  to  the  nimiber  of  periods. 

Note. — If  tlie  number  of  figures  in  the  given  number  is  odd,  the  left 
Jiand  period  will  have  but  one  figure. 

732.  1.  Required  the  length  of  one  side  of  a  square  garden 
which  contains  16  sq.  rods. 

Illustkation. — Let  the  garden  be  repre- 
sented by  the  adjoining  diagram. 

Now  as  the  garden  is  square,  its  sides  are 
equal,  and  the  length  of  one  side  is  one  of  the 
two  equal  factors,  or  the  square  root  of  16. 
But  16  =  4  X  4.  Hence,  the  length  of  a  side 
is  4  rods. 

Proof.— 4  rd.  x  4  rd.  =  16  sq.  rods,  the 

given  area.  a     a     -ia  a 

°  4  X  4  =  16  pq.  rods. 

2.  What  is  the  length  of  one  side  of  a  square  which  contains 
625  square  feet  ? 

OPERATION. 


4  ro 

ds. 

o 

■^ 

625  )  25 

45  )  225 
225 


Analysis. — Since  625  contains  three  figures,  it  must 
liave  two  i^eriods  ;  its  square  root  tw'o  figures,  and  first 
period  on  the  left  one  figure. 

The  greatest  square  of  6  (hundreds)  the  left  hand 
period  is  4  (hundreds)  and  its  root  is  2  (tens)  which  we 
place  on  the  right  for  the  first  figure  of  the  root.  Sub- 
tracting the  square  of  2  from  the  period  used,  we  annex 
to  the  remainder  the  next  period  for  a  dividend. 

Since  the  additions  are  to  be  made  on  two  sides  of  the  square,  we  place 
4,  the  double  of  tlie  root,  on  the  left  of  the  dividend  for  a  trial  divisor,  and 
find  it  is  contained  in  22,  5  times,  the  right  hand  figure  being  omitted. 
Placing  the  o  on  the  right  of  the  root  and  of  the  trial  divisor,  we  multiply 
the  divisor  thus  increased  by  this  figure,  and  subtracting  the  product  there 
is  no  remainder.     The  square  root  or  answer,  is  25  feet. 


Square  Root. 


295 


yO  feet. 


5  leet. 


Geometrical     Illustration. 

733.  1.  Take  any  number  as  625  sq.  ft.,  the  square  root  of 
which  is  to  be  found. 

Let  the  shaded  part  of  the  diagram 
represent  the  square  of  2  tens,  the  first 
figure  of  the  root ;  then  20  x  20,  or  400  sq. 
ft.,  will  be  its  contents.  Subtracting  the 
contents  from  the  given  area,  we  have 
625-400  =  225  sq.  ft.  to  be  added  to  this 
square.  To  preserve  its  form,  the  addition 
must  be  made  equally  to  two  adjacent 
sides.  The  question  is,  what  is  the  width 
of  the  addition. 

Since  the  length  of  the  square  is  20 
ft.,  adding  a  strip  1  foot  wide  to  two  sides 
will  take  20  +  20  or  40  sq.  ft.  Now  if  40  sq.  ft.  will  add  a  strip  1  foot  wide 
to  the  square,  225  sq.  ft.  will  add  a  strip  as  many  ft.  wide  as  40  is  contained 
times  in  225  ;  and  40  is  contained  in  225,  5  times  and  25  over. 

That  is,  since  the  addition  is  to  be  made  on  two  sides,  we  double  the 
root  or  length  of  the  side  found  for  a  trial  divisor,  and  find  it  is  contained 
in  225,  5  times,  which  shows  the  width  of  the  addition  to  be  5  feet. 

Now  the  length  of  each  side  addition  being  20  ft.,  and  the  width  5  ft., 
the  area  of  both  equals  20  x  5  +  20  x  5,  or  40  x  5  =  200  sq.  feet.  But  there 
is  a  vacancy  at  the  upper  corner  on  the  right,  whose  length  and  breadth 
are  5  ft.  each  ;  hence  its  area  =  5x5,  or  25  sq.  feet ;  and  200  sq.  ft.  +  25 
sq.  ft.  ■=  225  sq.  ft. 

For  the  sake  of  finding  the  area  of  the  two  side  additions  and  that  of  the 
corner  at  the  same  time,  we  place  the  quotient  5  on  the  right  of  the  root 
already  found,  and  also  on  the  right  of  the  trial  divisor  to  complete  it. 
Multiplying  the  divisor  thiis  completed  by  5,.  the  figure  last  placed  in  the 
root,  we  have  45  x  5  =  225  sq.  ft.  Subtracting  this  product  from  the  divi- 
dend, nothing  remains. 


2.  What  is  the  square  root  of  .576  ? 
Solution.— y^ySTG  =  a/,5766  =  .75  +  ,  Ans, 

3.  Find  the  sq.  root  of  234.09. 


Solution.— V234.O9  =  15.3,  Ans.     Hence, 


296  Evolution, 

734.  To  extract  the  square  root  we  have  the  following 

General     Rule. 

1.  Separate  the  number  into  periods  of  two  figures 
each,  beginning  at  units,  and  count  both  ways. 

IT.  Find  the  greatest  square  in  the  first  period  on  the 
left,  and'  place  its  root  on  the  right.  Subtract  this 
square  from  the  period,  and  on  the  right  of  the  remain- 
der place  the  next  period  for  a  dividend. 

III.  Double  the  part  of  the  root  thus  found  for  a  trial 
divisor ;  and  finding  how  many  times  it  is  contained  in 
the  dividend,  omitting  the  right  hand  figure,  annex  the 
quotient  both  to  the  root  and  to  the  divisor. 

IV.  Multiply  the  divisor  thus  increased  by  the  last 
figure  placed  in  the  root,  subtract  the  product  from  the 
dividend,  and  place  the  next  period  on  the  right  of  the 
remainder. 

V.  Proceed  as  before,  till  the  root  of  all  the  periods 
is  found. 

V^oo^.— Multiply  the  root  by  itself.     (Art.  722.) 

Notes. — 1.  If  tliere  is  a  remainder  after  the  root  of  the  last  period  is 
found,  annex  periods  of  ciphers,  and  proceed  as  before.  The  figures  of  the 
root  thus  obtained  will  be  decimals. 

2.  If  the  trial  divisor  is  not  contained  in  the  dividend,  annex  a  cipher 
both  to  the  root  and  to  the  divisor,  and  bring  down  the  next  period. 

3.  It  sometimes  happens  that  the  remainder  is  larger  than  the  divisor; 
but  it  does  not  necessarily  follow  that  the  figure  in  the  root  is  too  small. 

4.  The  left  ?innd  period  in  ^chole  numhcrs  may  have  but  one  figure ; 
but  in  deciinals,  each  period  must  have  two  figures.  Hence,  if  the  number 
of  decimals  is  odd,  a  cipher  must  be  annexed  to  complete  the  period. 

Find  the  square  root  of  the  following  numbers  : 


4. 

576. 

9. 

538.245. 

14. 

287.65. 

5. 

1600. 

10. 

61.7646. 

15. 

.776961. 

6. 

1225. 

11. 

8476.124. 

16. 

1073.741824, 

7. 

291.64. 

12. 

1232136. 

17. 

.00053361. 

8. 

864.91. 

13. 

5314491. 

18. 

617230.2096. 

Sqiiare  Root,  297 

735.  To  find  the  Square  Root  of  Fractioiw. 

1.  What  is  the  square  root  of  f|^  ? 

Solution. — \'\\  =  VtV  =  f»  ^^ns.     Hence,  the 

EuLE. — Reduce  the  fraction  to  its  simplest  form  and 
find  the  square  root  of  each  term  separately. 

Notes. — 1.  If  either  term  of  the  given  fraction,  when  reduced,  is  an 
imperfect  square,  reduce  tlie  fraction  to  a  decimal,  and  proceed  as  above. 
(Art.  249.) 

2.  Mixed  numbers  should  be  reduced  to  improper  fractions,  or  the 
fractional  part  to  a  decimal. 

2,   What  is  the  square  root  of  ^^^  ?     Ans.  |. 

Find  the  square  root  of  the  following  fractions  : 

^'        TT6*  °*        46T56-  '•         65§^36' 

4  2  5  6  6  1024  8  2  ft  1  6 

9.  What  is  the  square  root  of  20 J  ? 
Solution.— y'20 J  =  -^/^  =  |»  or  4^,  Ans. 
Find  the  square  root  of  the  following  : 


10. 

18-2. 

12. 

52^. 

14. 

II  of  144, 

11. 

40|f. 

13. 

113ff. 

15. 

16.  What  is  the  square  root  of  -g^  of  |f  oij^^  of  4096  ? 

17.  Required  the  square  root  of  3  to  7  decimals. 

18.  Required  the  square  root  of  12  to  eight  decimals. 

A.PPI^ICATIONS, 

735,  a.    1.  What  is  the  side  of  a  square  whose  area  contains 
2025  sq.  yards  ? 

2.  A  general  has  906304  soldiers;  how  many  must  he  place 
in  rank  and  file  to  form  them  into  a  square  ? 

3.  A  man  bought  a  square  tract  of  land  containing  3840 
acres  ;  how  many  rods  square  is  the  tract  ? 


298  Evolution. 

4.  What  is  the  side  of  a  square,  whose  area  is  equal  to  that 
of  a  triangle  containing  5184  sq.  ft.  ? 

5.  What  is  the  side  of  a  square  equal  in  area  to  a  rectangu- 
lar field  32  rods  long  and  18  rods  wide  ? 

6.  A  landholder  divided  a  tract  of  3802J  A.  into  four  equal 
and  square  farms  ;  what  is  the  length  of  one  of  their  sides  ? 

7.  A  man  having  a  garden  465  yards  square,  wished  to 
extend  it  so  as  to  make  it  9  times  as  large  ;  how  many  3'ards 
square  will  it  then  be  ? 

736.  A  mean  proportio7ial  between  tivo  ^uwihers  is  found  hy 
extracting  the  square  root  of  their  jyroduct.     (Art.  678.) 

8.  What  is  the  mean  proportional  between  9  and  16  ? 

Solution.— 16  x  9  =  144 ;  and  y^144  =  12,  Ans.    Hence, 

Note. — The  product  of  any  square  number  by  another  square  number 
is  always  itself  a  square. 

Find  the  mean  proportional  between  the  following  numbers: 


9. 

4  and  16. 

14. 

28  and  54. 

19. 

i  and  -J-. 

10. 

9  and  25. 

15. 

45  and  96. 

20. 

f  and  if  • 

11. 

25  and  36. 

16. 

.04  and  .16. 

21. 

If  and  %\. 

12. 

49  and  64. 

17. 

.64  and  6.25. 

22. 

If  and  ^V 

13. 

81  and  64. 

18. 

.09  and  .36. 

23. 

1  2  1    ^^^^    14  4  • 

737.  When  the  length  of  a  rectangular  field  equal  to  a  given 
area,  is  double,  triple,  etc.,  its  width,  its  dimensions  are  found 
by  extracting  the  square  root  of  \,  -j,  etc.,  of  the  area,  as  the 
case  may  be.  This  root  will  be  the  width,  and  being  doubled, 
tripled,  etc.,  wdll  be  the  length. 

24.  The  length  of  a  rectangular  field  containing  80  acres,  is 
twice  its  breadth  ;  what  are  its  length  and  breadth  ? 

25.  The  breadth  of  a  rectangular  farm  containing  160  acres, 
is  J  its  length  ;  what  are  its  length  and  breadth  ? 

738.  A  Triangle  is  a  figure  having  three  sides 
and  three  a^igles. 

739.  A  Eight-angled  Triangle  is  one  which 
has  a  right  angle  ;  as  ABC.  ^ 


Square  Root. 


299 


740.  The  Hypothenuse  of  a  right-angled  triangle  is  the  side 
AC,  opposite  the  right  angle  B ;  the  base  is  AB,  the  perpen- 
dicular is  BO. 

741.  The  relation  of  the  sides  of  a  tri- 
angle to  each  other  may  be  illustrated  as 
follows  : 

Take  any  right-angled  triangle  as  ABC, 
the  base  of  which  is  4  ft.,  the  perpendicu- 
lar 3  ft.,  and  the  hypothenuse  5  ft. 

It  will  be  seen  that  the  square  of  the  base 
contains  16  sq.  ft.,  that  of  the  perpendicu- 
lar 9  sq.  ft.,  and  that  of  the  hypothenuse 
25  sq.  ft.    Now  25  =  16  +  9.     Hence  we  derive  the  following 


Principles. 

i°.  Tlie  sum  of  the  squares  of  the  Base  and  Perpendicular 
is  equal  to  the  square  of  the  Hypothenuse. 

2°.  The  square  of  the  Hypothenuse  diminished  hy  the  square 
of  the  Perpendicular,  is  equal  to  the  square  of  the  Base. 

3°.  The  square  of  the  Hypothenuse  diminished  ly  the  square 
of  the  Base,  is  equal  to  the  square  of  the  Perpendicular. 
That  is, 

742.  The  square  descrihed  on  the  hypothenuse  of  a  right- 
angled  triangle  is  equal  to  the  sum  of  the  squares  of  the  base 
and  peiyendicular. 

743.  To  find  the  Hypothenuse,  when  the  Base  and  Perpendicular 

are  given. 

26.  What  is  the  length  of  a  ladder  which  will  just  reach  to 
the  top  of  a  house  32  feet  high,  when  its  foot  is  placed  24  feet 
from  the  house  ? 

Solution.— Perpendicular  (33)2  -  32  x  32  =  1024 
Base  (24)2  -  04  x  24  =    576 

The  square  root  of  tlieir  sum,      1600  =  40  ft.,  Ans. 


300  Evolution, 

Hence  we  have  the  following 

EuLE. — Add  the  square  of  the  base  to  the  square  of  the 
perpendicular,  and  the  square  root  of  the  sum  will  he  the 
hypothenuse. 


Formula. — HyiMlmiuse  =  VBase^  +  Perpendicular^ 

27.  The  side  of  a  certain  school-room  having  square  corners, 
is  8  yards,  and  its  width  6  yards ;  wliat  is  the  distance  between 
two  of  its  diagonal  corners  ? 

28.  Two  men  start  from  the  same  place  and  at  the  same 
time  ;  one  goes  exactly  south  40  miles  a  day,  the  other  goes 
exactly  west  30  miles  a  day  ;  how  far  apart  will  they  be  at  the 
close  of  the  first  day  ? 

29.  How  far  apart  will  the  same  travelers  be  in  4  days  ? 

744.  To  find  the  Perpendicular,  when  the  Base  and  Hypothenuse 

are  given. 

30.  A  line  10  yd.  long  fastened  to  the  top  of  a  tree,  reaches 
the  ground  6  yd.  from  the  base ;  what  is  the  height  of  the 
tree  ? 

Solution.— Hypothenuse  (10  yd.)^  =  10  x  10  =  100 
Base  (6yd.)2    =    6x6    =:    36 

Ttie  square  root  of  their  difference,  64  =;  8  yd.,  Ans. 
Hence,  the 

Rule. — From  the  square  of  the  hypothenuse  subtract 
the  square  of  the  base,  and  the  square  root  of  the  remain- 
der will  be  the  perpendicular. 

Formula. — Perpendicular  =  ^Hypothenuse^  —  Bas^. 

31.  A  line  75  feet  long  fastened  to  the  top  of  a  flag-staff 
reaches  the  ground  45  feet  from  its  base;  what  is  the  height 
of  the  flag-staff  ? 

32.  A  house  is  40  ft.  wide  and  the  length  of  the  rafters  is 
32  ft. ;  what  is  the  perpendicular  distance  from  the  beam  to 
the  ridgepole  ? 

33.  The  distance  between  the  diagonal  corners  of  a  croquet 
ground  is  17  yards,  and  its  length  is  15  yards ;  what  is  its  width  ? 


Square  Moot.  301 

745.  To  find  the  Base,  when  the  Hypothenuse  and  Perpendicular 

are  given. 

34.  A  ladder  50  ft.  long  was  placed  against  the  top  of  a 
house  40  ft.  high ;  what  distance  was  the  foot  of  the  ladder 
from  the  house  ? 

Solution.— Hypothenuse  (50  ft.)2  =  50  x  50  =  2500 
Perpendicular  (40  ft.)2  =  40  x  40  =:  1600 

The  square  root  of  their  difference,  900  =  30  ft.,  Ans. 
Hence,  the 

Rule. — From  the  square  of  the  hypothenuse  subtract 
the  square  of  the  jjerpeiidieular,  and  the  square  root  of 
the  reviainder  will  he  the  base. 


Formula. — Base  =  ^/  Hy2)othenuse^  —  Perpendicular'^. 

35.  The  slant  height  of  a  square  pyramid  is  40  ft.,  and  its 
perpendicular  height  32  ft.,  what  is  the  distance  from  the 
center  of  the  base  to  its  side  ? 

36.  The  height  of  a  tree  on  the  bank  of  a  river  is  100  ft, 
and  a  line  stretching  from  its  top  to  the  opposite  side  is 
144  ft.  ;  what  is  the  width  of  tlie  river  ? 

37.  The  side  of  a  square  field  is  30  rods ;  how  far  is  it  be- 
tween its  diagonal  corners  ? 

38.  If  a  square  field  contains  10  acres,  what  is  the  length  of 
its  side,  and  how  far  apart  are  its  diagonal  corners  ? 

39.  If  a  school  room  is  40  feet  long,  30  feet  wide,  and  14 
feet  high,  what  is  the  length  of  a  diagonal  drawn  upon  the 
floor ;  and  what  is  the  length  of  a  diagonal  drawn  from  the 
floor  to  the  ceiling  ? 

40.  A  park  53  rods  long  and  39  rods  wide  has  a  straight 
walk  running  from  its  diagonal  corners  ;  what  is  the  length  of 
the  walk  ? 

41.  The  side  of  a  square  room  is  40  feet ;  what  is  the  dis- 
tance between  its  diagonal  corners  on  the 'floor? 

42.  A  tree  was  broken  35  feet  from  its  root,  and  struck 
the  ground  21  ft.  from  its  base ;  what  was  the  height  of 
the  tree  ? 


302  Mvolution. 


Similar    Plane    Figures. 

746.  Similar  Plane  Figures  are  tliose  whicli  have  the  same 

form,  and  their  like  dimeiisions  proportional . 

Notes. — 1.  All  circles  and  all  rectilinear  figures  are  similar,  when  their 
several  angles  are  equal  each  to  each,  and  their  UTce  dimensions  propor- 
tional. 

2.  The  like  dimensions  of  circles  are  their  diameters,  radii,  and  circum- 
ferences. 

747.  The  Areas  of  similar  surfaces  are  to  each  other  as  the 
squares  of  their  like  dimensions.     Conyersely, 

The  Like  Dimensions  of  similar  surfaces  are  to  each  other  as 
the  square  roots  of  their  areas. 

1.  If  one  side  of  a  triangle  is  12  yards,  and  its  area  36  square 
yards,  what  is  the  area  of  a  similar  triangle,  the  corresponding 
side  of  which  is  8  yards  ? 

Solution.— (12)-^ :  (S)^  : :  36  :  Ans.,  or  16  sq.  yards. 

2.  If  one  side  of  a  triangle  containing  36  sq.  yards  is  8  yards, 
what  is  the  length  of  a  corresponding  side  of  a  similar  triangle 
which  contains  81  sq.  yards  ? 

Solution. — /y/36  :  y^Sl  : :  8  :  Ans.,  or  12  yards. 

3.  If  a  pipe  1  inch  in  diameter  will  fill  a  cistern  in  60  min., 
in  what  time  will  a  pipe  2  in.  in  diameter  fill  it  ? 

4.  If  a  gate  9  inches  in  diameter  will  empty  a  mill-pond  in 
16  hours,  how  large  must  a  gate  be  to  empty  it  in  4  hours  ? 

5.  If  one  side  of  the  base  of  a  triangular  pyramid  measuring 
16  square  feet,  is  20  inches  in  length,  what  is  the  length  of  a 
side  of  a  similar  pyramid,  which  measures  36  square  feet  ? 

6.  A  man  owns  a  building  lot  containing  20  square  rods  in 
the  shape  of  a  right-angled  triangle,  the  perpendicular  of  whicli 
is  20  yards  in  length  ;  what  is  the  perpendicular  of  a  similar 
lot,  which  contains  30  square  rods  ? 


Cube  Root  303 

Cube    Root. 

Oral     Exercises. 

748.  1.  What  number  taken  three  times  as  a  factor  pro- 
duces 8  ?     27  ? 

2.  What  is  one  of  the  three  equal  factors  of  64  ? 

3.  Name  one  of  the  three  equal  factors  of  125  ? 

4.  Name  one  of  the  three  equal  factors  of  1000.    Of  1728. 

Written     Exercises. 

749.  The  Cube  Root  of  a  number  is  one  of  its  three  equal 
factors. 

750.  To  find  the  number  of  figures  In  the  Cube  of  a   Number, 

also  in  the  Cube  Hoot  of  a  Number. 

1st.  Take  1  and  9,  also  10  and  99,  100  and  999,  etc.,  the  least  and  great- 
est integers  tliat  can  be  expressed  by  one,  tiDo,  three,  etc.,  figures. 

3d,  In  like  manner  take  .1  and  .9,  also  .01  and  .99,  etc.,  the  least  and 
greatest  decimals  that  can  be  expressed  by  one,  two,  etc.,  decimal  figures. 
Cubing  these,  we  have 


Roots. 

Powers. 

Roots. 

Powers. 

1 

13   =   1, 

.1 

.13  =  .001 

9 

93  =  729, 

.9 

.93  =  .729 

10 

103  =  1000, 

.01 

.013  ^  .000001 

99 

993  =  970299, 

.99 

.993  =  .970299 

100 

1003  =  1000000, 

.001 

.0013  ^  .000000001 

999 

9993  =  997002999, 

.999 

.9993  =  .997002999 

omparir 

ig  these  roots  and  their 

cubes,  we 

discover  the  followi 

PrI  N  CI 

PLES. 

1°.  The  cube  of  a  number  cannot  have  more  thati  three  times 
as  many  figures  as  its  root,  nor  but  two  less. 

28.  If  a  number  is  separated  into  periods  of  three  figures  each 
beginning  at  units  place,  the  number  of  figures  in  the  cube  root 
ivill  be  the  same  as  the  number  of  ijeriods. 


304  Evolution. 

Notes. — The  left  hand  period  in  \diole  numbers  may  be  incompletCf 
having  only  one  or  two  figures;  but  each  period  of  decimals  must  always 
have  three  figures.  Hence,  if  the  decimal  figures  in  a  given  number  are 
less  than  three,  annex  ciphers  to  complete  the  period. 

How  many  figures  in  the  cube  root  of  the  following : 


1. 

340566. 

3. 

576.453. 

5. 

32.7561o 

2. 

1467. 

4. 

5.7321. 

6. 

.456785. 

751.  To  find  the  Cube  of  a  number  consisting  of  two  figures  in 
the  terms  of  its  parts. 

1.  Find  the  cube  of  35  in  the  terms  of  its  tens  and  units. 


35  = 
35  = 

OPERATION. 

30  +  5 
30  +  5 

175  = 
105    = 

(30x5)  +52 
302+   (30x5) 

1225   =: 

35  .= 

302  +  2(30x5)  +52 
30  +  5 

6125  = 
3675     := 

(302 
30-^  +  2(302 

X5)  +2(30x52) +  53 
X  5)  +   (30  X  52) 

42875  =  303 +  3(302x5) +  3(30x52) +  53. 

Explanation.— The  cube  of  the  tens,        (SO^)  =  27000 

3  times  the  square  of  tens  by  units,   3  (30^  x  5)  =  13500 

3  times  the  tens  by  square  of  units,  3  (80  x  5'^)  =     2250 

and  the  cube  of  the  units  5^  =125 

Now  27000  +  13500  +  2250  +  125  =  42875.     Hence, 

752.  The  cube  of  any  numher  consisting  of  tens  and  units  is 
equal  to  the  cuhe  of  the  tens,  plus  3  times  the  square  of  the  tens 
dy  the  units,  plus  3  times  the  tens  hy  the  square  of  the  units, 
plus  the  cube  of  the  units. 

Note.  — Since  the  cube  of  a  number  consisting  of  tens  and  imits  is  equal 
to  the  cube  of  the  tens,  plus  3  times  the  square  of  the  tens  by  the  units, 
etc.,  when  a  number  has  two  jieriods,  it  follows  that  the  left  hand  period 
must  contain  the  cube  of  the  tens,  or  first  figure  of  the  root. 

2.  Find  the  cube  of  32  in  the  terms  of  its  tens  and  units. 


Cube  Root 


305 


753.  To  Extract  the  Cube  Root  of  a  number. 

1.  What  is  the  side  of  a  cube  which  contains  27  solid  feet  ? 


Illustration. — Let  the  cube  be  represented 
by  the  adjoming  diagram,  each  side  of  which  is 
divided  into  9  square  feet.  Since  the  length  of 
a  side  is  3  feet,  if  we  multiply  3  into  3  into  3, 
the  product  27,  will  be  the  solid  contents  of  the 
cube.  (Art.  429.)  Now,  if  we  reverse  the  pro- 
cess, dividing  37  into  three  equal  factors,  one  of 
these  factors  will  be  the  side  of  the  cube. 
Ans.  3  ft. 


3  feet. 


,...: ^ 


'I         ■  i 

I,         !             \ 
Ii  '  "■■ i 


3  X  3  X  3  =  27  ft. 


2.  What  is  the  length  of  one  side  of  a  cubical  mound  con- 
taining 15625  solid  feet  of  earth  ? 


OPERATION. 


15625  (  25 


1200 

300 

_25^ 

1525 


7625 


7625 


Explanation. — 1.  We  separate  the  given  num- 
ber into  periods  of  three  figures  each,  placing  a 
point  over  units,  then  over  thousands.  This  shows 
that  the  root  must  have  two  figures. 

3.  Beginning  with  the  first  period  on  the  left, 
we  find  the  greatest  cube  in  15  is  8,  the  root  of 
which  is  3.  Placing  the  3  on  the  right,  we  sub- 
tract its  cube  from  the  period,  and  to  the  remain- 
der bring  down  the  next  period  for  a  dividend. 
This  shows  that  we  have  7635  solid  feet  to  be  added. 

3.  We  square  the  root  already  found,  which  in  reality,  as  there  is  to  be 
another  figure  in  the  root,  is  30 ;  then  multiplying  its  square  400  by  3,  we 
write  the  product  on  the  left  of  the  dividend  for  a  trial  divisor  ;  and  find- 
ing it  is  contained  in  the  dividend  5  times,  place  the  5  in  the  root. 

4.  We  next  multiply  30,  the  root  already  found  by  5,  the  last  root 
figure ;  then  multiply  this  product  by  3  and  write  it  under  the  divisor. 
We  also  write  the  square  of  5,  the  last  figure  placed  in  the  root,  under 
the  divisor.  Adding  these  three  results  together,  multiply  their  sum 
1535  by  5,  and  subtract  the  product  from  the  dividend.  The  answer 
is  35. 


Tllustratiox    by     Cubical    I^locks* 

Let  the  adjoining  diagram  represent  a  set  of  cubical  blocks.  Let  the 
cube  of  30,  the  tens  of  the  root,  be  represented  by  the  large  cube.  The 
remainder  7635  is  to  be  added  equally  to  three  adjacent  sides  of  this  cube. 


*  Every  gchool  in  which  cube  root  is  taught,  should  be  furnished  with  a  set  of 
Cubiral  Blocks, 


306 


Evolution. 


20  ft. 


5  It. 


To  ascertain  the  thickness  of  these  side  20  ft. 

additions,  we  form  a  trial  divisor  by  squar- 
ing 2,  the  first  figure  of  the  root,  with  a 
cipher  annexed,  for  the  area  of  one  side  of 
this  cube,  and  multiply  this  square  by  3 
for  the  three  side  additions.  Now  20'^  =  20 
X  20  =  400  ;  and  400  x  3  =  1200,  the  trial 
divisor.  Dividing  7625  by  1200,  the  quo- 
tient 5,  shows  that  the  side  additions  are  to 
be  5  ft.  thick,  and  is  placed  on  the  right  for 
the  units'  figure  of  the  root. 

To  represent  these  additions,  place  the  corresponding  layers  on  the  top, 
front,  and  right  of  the  large  cube.  But  we  discover  three  vacancies  along 
the  edges  of  the  large  cube,  each  of  which  is  20  ft.  long,  5  ft.  wide,  and 
5  ft.  thick.  Filling  these  vacancies  with  the  corresponding  rectangular 
blocks,  we  discover  another  vacancy  at  the  junction  of  the  corners  just 
filled,  whose  length,  breadth,  and  thickness  are  each  5  ft.  This  is  filled 
by  the  small  cube. 

To  complete  the  trial  divisor,  we  add  the  area  of  one  side  of  each  of  the 
corner  additions,  viz.,  20  x  5  x  3,  or  300  sq.  ft.,  also  the  area  of  one  side  of 
the  small  cube  =  5x5,  or  25  sq.  ft.  Now  1200  +  300+25  =  1525.  The 
divisor  is  now  composed  of  the  area  of  3  sides  of  the  large  cube,  plus  the 
area  of  one  side  of  each  of  the  corner  additions,  plus  the  area  of  one  side 
of  the  small  cube,  and  is  complete. 

To  ascertain  the  contents  of  the  several  additions,  we  multiply  the 
divisor  thus  completed  by  5,  the  last  figure  of  the  root ;  and  1525  x  5  —  7625. 
Subtracting  the  product  from  the  dividend,  nothing  remains.     Hence, 

754.  To  extract  the  cube  root  we  have  the  following 


General     Rule. 

I.  Separate  the  given  numher  into  periods  of  three 
figures  each;  begin  luith  units  and  count  hoth  ways. 

II.  Find  the  greatest  cube  in  the  first  period  on  the 
left,  andy  place  its  root  on  the  riglvt.  Subtract  this  cube 
froirv  the  period,  and  to  the  right  of  the  remainder 
bring  down  the  next  period,  for  a  dividend. 

III.  Multiply  the  square  of  the  root  thus  found,  con- 
sidered as  tens,  by  three,  for  a  trial  divisor;  and 
finding  how  many  times  it  is  contained  in  the  divi- 
dend, write  the  quotient  for  the  second  figure  of  the  root. 


Cube  Root,  307 

IV.  To  complete  the  trial  divisor,  add  to  it  three 
times  the  product  of  the  root  previously  found  ivith  a 
cipher  annexed,  by  the  second  root  figure,  also  add  the 
square  of  this  second  figure. 

V.  Multiply  the  divisor  thus  completed  hy  the  last 
figure  placed  in  the  root.  Subtract  the  product  from 
the  dividend;  and  to  the  right  of  the  remainder  bring 
down  the  next  period  for  a  new  dividend.  Find  a  neiv 
trial  divisor  as  before,  and  thus  proceed  till  the  root 
of  the  last  period  is  found. 

Notes. — 1.  If  there  is  Si  remainder  after  tlie  root  of  tlie  last  period  is 
found,  annex  periods  of  ciphers,  and  proceed  as  before.  The  root  figures 
thus  obtained  will  be  decimals. 

2.  If  a  trial  divisor  is  not  contained  in  the  dividend,  put  a  cipher  in 
the  root,  two  ciphers  on  the  right  of  the  divisor,  and  bring  down  the  next 
period. 

3.  If  the  product  of  the  di\'isor  completed  into  the  figure  last  placed  in 
the  root  exceeds  the  dividend,  the  root  figure  is  too  large.  Sometimes  the 
remainder  is  larger  than  the  divisor  completed  ;  but  it  does  not  necessa- 
rily follow  that  the  root  figure  is  too  small. 

3.  What  is  the  cube  root  of  130241.7  ? 

Explanation.  —  Having  completed 
the  period  of  decimals  by  annexing  two 
ciphers,  we  find  the  first  figure  of  the 
root  as  above.  We  place  the  next 
period  on  the  right  of  the  remainder, 
and  the  dividend  is  5241.  The  trial 
divisor  7500  is  not  contained  in  the 
dividend ;  therefore,  placing  a  cipher  in 
the  root  and  two  ciphers  on  the  right  of 
the  divisor,  we  bring  down  the  next 
period ,  and  proceed  as  before. 

Extract  the  cube  root  of  the  following  numbers  ; 


136241.700(50.6  + 
125 

750000 

5241.700 

9000 

36 
759036 

4554216 

687484  Eem. 

4. 

13824. 

8. 

1092727. 

12. 

91.125. 

5. 

571787. 

9. 

2357947691. 

13. 

.253395799. 

6. 

373248. 

10. 

27054036008. 

14. 

164.566592. 

7. 

1953125. 

11. 

12.167. 

15. 

122615.327232. 

308  Evolution. 

755.  To  find  the  cube  root  of  a  common  fraction,  reduce  the 
fraction  to  its  loioest  terms,  then  extract  the  root  of  its  7i2imera- 
tor  a?id  denominator. 

Notes. — 1.  When  either  the  numerator  or  denominator  is  not  2i  perfect 
cube,  the  fraction  should  be  reduced  to  a  decimal,  and  the  root  of  the  deci- 
mal be  found  as  above. 

2.  A  mixed  number  should  be  reduced  to  an  improper  fraction. 

16.  What  is  the  cube  root  of  ^y^  ? 
Solution.— ^^  =  ^f|  =  f ,  Ans. 
Find  the  cube  root  of  the  following : 

1  n  3  T6_  10         1520  91         iq2 

23.  Find  the  cube  root  of  2  to  4  places  of  decimals. 

24.  Find  the  cube  root  of  3  to  5  places  of  decimals. 

Applic.4^tions. 

756.  1.  What  is  the  length  of  a  side  of  a  cubical  box,  which 
contains  389017  solid  inches? 

2.  Find  the  side  of  a  cu.  vat,  which  contains  48228544  cu.  feet? 

3.  What  is  the  side  of  a  cubical  mound,  which  contains 
1259712  solid  yards  ? 

4.  What  is  the  side  of  a  cube  equal  to  a  stick  of  timber  2 
feet  square  and  128  feet  long  ? 

5.  What  is  the  side  of  a  cubical  bin,  which  contains  500 
bushels,  allowing  2150.4  cu.  in.  to  a  bushel  ? 

6.  What  is  the  side  of  a  cubical  cistern,  which  holds  100 
wine  hogsheads  ? 

7.  What  is  the  side  of  a  cube  equal  to  a  pile  of  wood  2421  ft 
long,  12  ft.  wide,  and  7  feet  high  ? 

Similar    Solids. 

757.  Similar  Solids  are  those  which  have  the  same  form,  and 
their  like  dimensions  proportional. 

Notes. — 1.  The  like  dimensions  of  spheres  are  their  diameters,  radii, 
and  circumferences  ;  those  of  cubes  are  their  sides. 


Cube  Root.  309 

2.  The  like  dimensions  of  cylinders  and  cones  are  their  altitudes,  and 
the  diameters  or  the  circumferences  of  their  bases. 

3.  Pyramids  are  similar,  when  their  bases  are  similar  polygons,  and 
their  altitudes  proportional. 

4.  Polyhedrons  {i.  e.,  solids  included  by  any  number  of  plane  faces)  are 
similar,  when  they  are  contained  by  the  sa7ne  nmnber  of  similar  polygons, 
and  all  their  solid  angles  are  equal  each  to  each. 

758.  The  Contents  of  similar  solids  are  to  each  other  as  the 
oubes  of  their  like  ditnensions.     Conversely, 

The  Like  Dimensions  of  similar  solids  are  as  the  cube  roots  of 
their  contents. 

1.  If  a  globe  4  inches  in  diameter  weighs  32  lbs.,  what  is  the 
weight  of  a  globe  whose  diameter  is  5  inches  ? 

Solution. — 4^ :  5^  :  :  32  lbs.  :  Ans. 

125  X  32  lbs.  =  4000  lbs.,  and  4000  lbs. -^64  =  62.5  lbs.,  Aris. 

2.  If  a  sphere  3  inches  in  diameter  weighs  4  lbs.,  what  is  the 
diameter  of  a  sphere  which  weighs  32  lbs.  ? 

Solution. — 4  lbs. :  32  lbs.  : :  3^ :  cube  of  diameter  required. 
Now  32  X  27  =  864;  then  864-T-4  =  216,  and  ^^216  =  6  in.,  Ans. 

3.  If  a  cannon  ball  6  inches  in  diameter  weighs  58  lbs.,  what 
is  the  weight  of  a  similar  ball  8  inches  in  diameter  ? 

4.  If  a  cube  of  gold  whose  side  is  3  inches  is  worth  $6400, 
what  is  the  worth  of  a  cube  of  gold  whose  side  is  8  inches  ? 

5.  If  a  pyramid  60  feet  high  contains  12500  en.  ft.,  how 
many  en.  ft.  are  there  in  a  similar  pyramid  30  ft.  high  ? 

6.  If  a  conical  stack  of  hay  whose  height  is  12  feet  contains 
5  tons,  what  is  the  weight  of  a  similar  stack  whose  height  is 
20  feet  ? 

7.  If  a  cubical  block  of  marble  whose  side  is  4  inches  weighs 
12  pounds,  what  will  a  cubic  foot  of  marble  weigh  ? 

8.  If  a  cylindrical  cistern  6  feet  in  diameter  will  contain  30 
hogsheads  of  water,  how  much  will  a  similar  cistern  contain, 
whose  diameter  is  20  feet  ? 


310  Evolution. 

759.  The  side  of  a  cube  whose  solidity  is  double,  triple,  etc., 
that  of  a  cube  whose  side  is  given,  is  found  by 

CuMng  the  given  side,  rrmltiplying  it  hy  the  given  proportion, 
and  extracting  the  cube  root  of  the  product, 

9.  What  is  the  side  of  a  cubical  mound,  which  contains  8 
times  as  many  solid  feet  as  one  whose  side  is  3  ft.   Ans.  6  ft. 

10.  Required  the  side  of  a  cubical  vat,  which  contains  3  times 
as  many  solid  feet  as  one  whose  side  is  5  ft. 

11.  If  a  cube  of  silver  whose  side  is  4  inches,  is  worth  $200, 
what  is  the  side  of  a  cube  of  silver,  worth  $1000? 

12.  I  have  a  cubical  box  whose  side  is  6  ft. ;  I  want  another 
which  will  contain  \  j^art  as  much.  What  will  be  the  length 
of  its  side  ? 

13.  Required  the  side  of  a  cubical  vat  which  shall  contain 
■^  part  as  much  as  one  whose  side  is  12  feet  ? 

Questions. 

710.  What  is  involution?  711.  What  is  a  power?  713.  The  first 
power?  714.  The  second  ?  715.  The  third  ?  716.  What  is  an  exponent  ? 
718.  How  find  a  power  of  a  number  ? 

731.  What  is  evolution?  722.  What  is  a  root?  723.  Square  root? 
724.  Cube  root  ?  727.  A  perfect  power  ?  728.  Imperfect  ?  729.  What  is 
extracting  the  square  root?  731.  Name  the  principles  respecting  squares 
and  root  ?  734.  How  extract  the  square  root.  735.  How  find  the  square 
root  of  fractions  ? 

736.  How  find  a  mean  proportional  between  two  numbers  ?  739.  What 
is  a  right-angled  triangle  ?  740.  Which  side  is  the  hypothenuse  ?  What 
are  the  other  two  sides  called  ?  741.  Name  the  principles  respecting 
right  angled  triangles.  742.  To  what  is  the  square  of  the  hypothenuse 
equal  ?  746.  What  are  similar  figures  ?  747.  How  do  similar  surfaces 
compare  with  each  other  ? 

749.  What  is  the  cube  root  of  a  number?  750.  Name  the  principles 
respecting  the  number  of  periods  and  figures  in  the  root  ?  752.  To  what 
is  the  cube  of  a  number  consisting  of  tens  and  units  equal?  754.  How 
extract  the  cube  root  ?     755.  How  find  the  cube  root  of  a  fraction  ? 

757.  What  are  similar  solids?  What  are  the  like  dimensions  of 
spheres?  Of  cubes?  Of  cylinders  and  cones?  Of  pyramids?  758. 
How  do  the  contents  of  similar  solids  compare  with  each  other  ? 


EOGKESSIOI^. 


Definitions. 

760.  A  Progression  is  a  series  of  numbers  which  regularly 
mcrease  or  decrease. 

761.  The  Terms  of  a  Progression  are  the  numbers  which 
form  the  series.  T\\q  first  and  last  terms  are  the  extremes  ;  the 
others,  the  means. 

762.  Progressions  are  of  two  kinds,  aritlunetical  and  geo- 
metrical. 

Arithmetical    Progression. 

763.  An  Arithmetical  Progression  is  a  series  w^hich  increases 
or  decreases  by  a  common  difference. 

764.  The  Common  Difference  of  a  progression  is  the  differ- 
ence between  any  two  of  its  consecutive  terms. 

765.  In  an  ascending  series,  each  term  is  found  hy  adding 
the  common  difference  to  the  preceding  term.     Thus, 

If  the  first  term  is  1  and  the  common  difference  3,  the  series  is 
1,    4,     7,     10,    13,     16,    19,    etc. 

766.  In  a  descending  series,  each  term  is  found  by  subtract- 
ing  the  common  difference  from  the  preceding  term.     Thus, 

If  15  is  the  first  term  and  2  the  common  difference,  the  series  is 

15,     13,     11,    9,     7,    5,     3,     1. 

Notes. — 1.  An  Arithmetical  Progression  is  sometimes  called  &n.Bqui- 
Mfferent  Series.  In  every  progression  there  may  be  an  infinite  number  of 
ferms. 

2.  An  Arithmetical  Mean  between  two  numbers  is  found  by  taking  half 
their  sum. 


312  Progression, 

IQl.  In  Arithmetical  Progression  there  are  five  elements  or 
paints  to  be  considered  :  the  first  term,  the  common  difference, 
the  last  term,  the  numher  of  terms,  and  the  sum  of  the  terms. 

Let  (I  =   the  first  term 

I  —  the  last  term. 
(I  =  the  common  difference. 
ti  =  the  number  of  terms. 
s  =  the  smn  of  the  terms. 

The  rehation  of  these  five  quantities  to  each  other  is  such 
that  if  any  three  of  them  are  given,  the  othe?'  tivo  can  be  found. 

768.  To  find  the  Last  Tei'm,  when  the  First  Term,  the  Common 

Difference,  and  Number  of  Terms  are  given. 

1.  Find  the  last  term  of  an  increasing  series  having  7  terms, 
its  first  term  being  3,  and  its  common  difference  2. 

Analysis. — From  the  definition,  each  succeeding  term  is  found  by  add- 
ing the  common  difference  to  the  preceding.     The  series  is : 

3,    3  +  2,    3  +  (2  +  3),    3  +  (2  +  2  +  2),     3  +  (2-f2  +  2  +  2),  etc.     Or, 
3,    3  +  2.    3  +  (2x2),     3  +  (2x3),  3  +  (2x4),  etc. 

2.  Find  the  last  term  of  a  decreasing  series  having  5  terms, 
the  first  term  being  24,  the  common  difference  2. 

Analysis. — In  a  descending  series,  each  succeeding  term  is  found  by 
subtracting  the  common  difference  from  the  preceding.  Hence,  the 
series  is 

24,     24-2,     24-(2  +  2),     24-(2  +  2  +  2),     24-(2  +  2  +  2  x2).  etc.    Or, 
24,     24-2,     24-(2x2),     24-(2  x  3),  24-(2  x4),  etc.     That  is, 

769.  The  last  term  is  equal  to  the  first  term,  increased  or 
diminished  by  the  product  of  the  common  difference  into  the 
number  of  terms  less  1.     Hence,  the 

Rule. — I.  Multiply  the  mnnher  of  teTins  less  one  by 
the  common  clifference. 

II.  WJien  the  series  is  ascending,  add  this  product  to 
the  first  term;  when  descending,  subtract  it  from  the 
first  term, 

FOKMULAS.-?  =  I  «  +  (»  -  J)  X  d.    Or, 

(  a  —  {n  —  1)  X  a. 


Arithmetical  Progression.  313 

770.  To  find  the  First  Term,  when  the  Last  Term,  the  Common 
Difference,  and  Number  of  Terms  are  given. 

1.  Find  the  first  term  of  a  decreasing  series  the  last  term  of 
which  is  2,  the  common  difference  3,  the  number  of  terms  6. 

Analysis. — The  first  term  of  a  decreasing  series  will  be  the  last  term 
increased  by  the  product  of  the  common  difference  by  the  number  of  terms 
less  one.     The  series  is 

3  +  3x5,    2  +  8x4.     2  +  3x3,     2  +  3x2,     2  +  3,    2. 

2.  Find  the  first  term  of  an  increasing  series,  the  last  term 
of  which  is  45,  the  common  difference  5,  and  the  number  of 
terms  7. 

Analysis. — The  first  term  of  an  increasing  series  will  be  the  last  term 
diminished  by  the  product  of  the  common  difference  by  the  number  of 
terms  less  one.     The  series  is 

45-5x6,    45-5x5,    45-5x4,     45-5x3,     45-5x2,     45-5x1,     45. 

Hence,  the 

Rule. — I.  Multiply  the  ninnher  of  terms  less  one  hy 
the  coimnon  difference. 

II.  When  the  series  is  ascending,  subtract  this  product 
from  the  last  term;  when  descending,  add  it  to  the 
last  term. 

^  \l  —  (n  —  1)  X  d.     Or, 

Formulas. — a  —  {  j      )  y, 

{  t  +  {ii  —  1)  X  d. 

Note. — Any  term  in  the  series  may  be  found  by  the  preceding  rules. 
For,  the  series  may  be  supposed  to  stop  at  any  term,  and  that  may  be 
considered  the  last. 

3.  Find  the  last  term  of  an  ascending  series,  the  first  term 
of  which  is  5,  the  common  difference  3,  and  the  number  of 
terms  12? 

4.  The  first  term  of  a  descending  series  is  40,  the  common 
difference  3,  and  the  number  of  terms  11  ;  what  is  the  last? 

5.  The  last  term  of  an  ascending  series  is  87,  the  number  of 
terms  16,  and  the  common  difference  4 ;  what  is  the  first  term  ? 

6.  What  is  the  amount  of  ^250,  at  Q%  simple  interest,  for 
21  years  ? 


314  Progression, 

ITL.  To  find  the  Number  of  Terms,  when  the  Extremes  and 
the  Common  Difference  are  given. 

1.  The  extremes  of  an  arithmetical  series  are  4  and  37,  and 
the  common  difference  3 ;  what  is  the  number  of  terms? 

Analysis. — The  last  term  of  a  series  is  equal  to  tlie  first  term  increased 
or  diminished  by  the  product  of  the  common  difference  by  the  number  of 
terms  less  1.     (Art.  769.) 

Now  37—4,  or  83,  is  the  product  of  the  common  difference  3,  by  the 
number  of  terms  less  1.  Consequently  33-J-3,  or  11,  must  be  the  number 
of  terms  less  1  ;   and  11  +  1,  or  12,  is  the  answer  required.     Hence,  the 

Rule. — Divide  the  difference  of  the  extremes  by  the 
coimnon  difference,  and  add  1  to  the  quotient. 

Formula. — n  =  \  —^ h  1, 

2.  The  age  of  the  youngest  child  of  a  family  is  1  year,  the 
oldest  22,  and  the  common  difference  of  their  ages  3  yr.;  how 
many  children  in  the  family  ? 

3.  The  extremes  of  an  arithmetical  series  are  8  and  96,  the 
common  difference  4  ;  what  is  the  number  of  terms  ? 

4.  A  laborer  worked  for  50  cts.  the  first  day,  54  cts.  the 
second,  58  cts.  the  third,  and  so  on  till  his  wages  were  12  a 
day ;  how  many  days  did  he  work  ? 

772.  To  find   the  Common  Difference,   when  the   Extremes 
and  the  Number  of  Terms  are  given. 

1.  The  extremes  of  a  series  are  3  and  21,  and  the  number  of 
terms  is  10  ;  what  is  the  common  difference  ? 

Analysis. — The  difference  of  the  extremes  21 — 3  =  18,  is  the  product 
of  the  number  of  terms  less  1  by  the  common  difference,  and  10—1,  or  9, 
is  the  number  of  terms  less  1 ;  therefore  18-5-9,  or  2,  is  the  common  differ- 
ence required.     (Art.  764.)     Hence,  the  . 

EuLE. — Divide  the  difference  of  the  extremes  hy  the 
nujnber  of  terms  less  1. 

Formula. — d  =  ^    '  ~      - 
n  —  1 


Aritlimetical  Progression,  315 

2.  The  ages  of  10  children  form  an  arithmetical  series  ;  the 
youngest  is  3  yr.  and  the  eldest  30  years ;  what  is  the  differ- 
ence of  their  ages  ? 

3.  A  military  company  appropriated  1108  for  8  target  prizes, 
the  highest  of  which  was  $24,  and  the  lowest  $3  ;  what  was 
the  common  difference  in  the  prizes  ? 

4.  The  amount  of  $600  for  45  yr.  at  simple  interest  is  $3120 ; 
what  is  the  rate  per  cent  ? 

5.  The  amount  of  $1500  for  27  years  is  $1620;  wdaat  is  the 
rate  per  cent  ? 

773.  To  find  the  Suin  of  all  the  terms,  when  the  Extremes  and 
the  Number  of  Terms  are  given. 

1.  Eequired  the  sum  of  the  series  having  7  terms,  the 
extremes  being  3  and  15. 

Analysis.— (1.)  The  series  is  3,     5,     7,     9,    11,   13,   15. 
(2.)  Inverting  the  same,     15,    13,    11,     9,     7, 5,     3. 

(3.)  Adding  (1.)  and  (2.),     18  +  18  +  18  +  18  +  18  +  18  +  18=twice  thesnm. 
(4.)  Dividing  (3.)  by  2,  9+  9+  9+   9+   9+   9+  9=63.  the  sum. 

By  inspecting  these  series,  we  discover  that  half  the  sum  of  the  extremes 
is  equal  to  the  average  value  of  the  terras.     Hence,  the 

EuLE. — Multiply  half  the  sinn  of  the  eoctr ernes  by  the 
ninnher  of  terms. 

Formula.— s  —  ]  — ^—  x  n. 

Note. — From  the  preceding  illustration  we  see  that, 

Th(}  sum  of  the  extremes  is  equal  to  the  sum  of  any  tim  terms  equidistant 
from  them  ;  or,  to  twice  the  sum  of  the  middle  term,  if  the  number  of 
terms  he  odd. 

2.  How  many  strokes  does  a  common  clock  strike  in 
12  hours? 

3.  Find  the  sum  of  all  the  terms,  the  extremes  being  0  and 
300,  and  the  number  of  terms  1200. 

4.  A  father  deposited  $1  in  the  bank  for  his  daughter  on 
her  first  birthday,  %^  the  next,  $7  the  next,  and  so  on;  how 
much  did  she  have  when  she  was  21  years  old  ? 


316  Progression, 

Geometrical    Progression. 

Definitions. 

774.  A  Geometrical  Progression  is  a  series  of  numbers 
which  increase  or  decrease  by  a  common  ratio. 

775.  The  Terms  of  a  geometrical  progression  are  the  num- 
bers which  form  the  series. 

Note. — The  series  is  called  Ascending  or  Descending,  according  as  the 
terms  increase  or  decrease.    (Arts.  763,  764.) 

776.  In  an  ascending  series  the  ratio  is  greater  than  one. 
Thus,  2,  4,  8,  16,  32,  64,  etc.,  is  an  ascending  progression. 

777.  In  a  descending  series  the  ratio  is  tess  than  one. 
Thus,  1, 1,  \,  I,  iV»  O'  6*^-'  is  ^  descending  progression. 

778.  In  Geometrical  Progression  there  are  also  five  elements 
or  parts  to  be  considered,  viz. :  i\\Q  first  term,  the  last  term,  the 
numher  of  terms,  the  ratio,  and  the  sum  of  all  the  terms. 

Let  ct  =  the  first  term. 

I  =  the  last  term. 

V  =  the  ratio. 

11  =  the  number  of  terms. 

s  =  the  sum  of  the  terms. 

779.  To  find  the  Last  Term,  when  the   First  Term,  the  Ratio, 

and  the  Number  of  Terms  are  given. 

1.  Required  the  last  term  of  an  ascending  series  having 
6  terms,  the  first  term  being  3,  and  the  ratio  2. 

Analysis. — From  the  definition,  the  series  is 

3,    3x2,    3x(2x2),     3x(2x2x2),    3  x  (2  x2  x2  x2),  etc.     Or, 

3,    3  X  2,    3  X  22,  3  X  23,  3  x  2^,  etc. 

Now,  3  X  25  =  3  X  32  =  96,  A  ns.    That  is, 

Each  successive  term  =  1st  term  x  ratio  raised  to  a  power  whose  ex- 
ponent is  one  less  than  the  number  of  th.e  term.     Hence,  the 


Geometrical  Progression.  317 

KuLE. — Multiply  the  first  term  hy  that  power  of  the 
ratio  whose  exponent  is  1  less  than  the  number  of  terms. 

Formula. — I  =  a  x  ^•»-^. 

Notes. — 1.  Any  term  in  a  series  may  be  found  by  tlie  preceding  rule. 
For,  tlie  series  may  be  supposed  to  stop  at  that  terra. 

2.  The  preceding  rule  is  applicable  to  Compound  Interest ;  the  principal 
being  the  first  term  of  the  series  ;  the  amount  of  $1  for  1  year  the  ratio ; 
and  the  number  of  year ><  plus  1,  the  number  of  terms. 

2.  A  father  promised  his  son  1  ct.  for  the  first  example  he 
solved,  2  cts.  for  the  second,  4  cts.  for  the  third,  etc. ;  what 
would  the  son  receive  for  the  tenth  example  ? 

3.  What  is  the  ami  of  $375  for  4  yr.,  at  b%  compound  int.  ? 

4.  "What  is  the  amount  of  11200  for  5  years,  at  Q'-c  comj^ound 
interest  ?     Of  $2500  for  4  years,  at  7^  ? 

780.  To   find    the    First   Term,  when    the   Last  Term,  the  Ratio, 

and  the  Number  of  Terms  are  given. 

1.  The  last  term  of  a  progression  is  96,  the  number  of  terms 
6,  and  the  ratio  2  ;  what  is  the  first  term  ? 

Analysis. — Reversing  the  steps  of  the  preceding  rule,  we  have  96-i-2^ 
=  96-V-33  =  3,  Ans.    Hence,  the 

EuLE. — Divide  the  last  term  by  that  power  of  the  ratio 
whose  exponent  is  1  less  than  the  number  of  terms. 

Formula. — a  =  I  -^  ?♦"-'. 

2.  The  last  term  of  a  series  is  192,  the  ratio  3,  and  the  num- 
ber of  terms  7  ;  what  is  the  first  term  ? 

781.  To  find  the  Stiin  of  all  fJie  Terms,  when  the  Extremes 

and  Ratio  are  given. 

1.  Required  the  sum  of  the  series  whose  first  and  last  terms 
are  2  and  162,  and  the  ratio  3. 

Analysis. — Since  each  succeeding  term  is  found  by  multiplying  the 
preceding  term  by  the  ratio,  the  series  is  2,  6, 18,  54,  162. 

(1. )  The  sum  of  the  series,         =  2  +  6  + 18  +  54  4- 162. 
(2.)  3  times  the  sum  =        6+18  +  54+162  +  486. 

Subt.  (1.)  from  (2.),  we  have      486—2  =  484,  or  twice  the  sum. 
Therefore,  484^2  =  242,  the  sum  required.    Hence,  the 


318  Progression, 

EuLE. — Multiply  the  last  term  dy  the  ratio,  and  sub- 
tracting the  first  terin  from  the  product,  divide  the 
remainder  by  the  ratio  less  1. 

-r^^  {I  X  r)  —  a 

FORMVLA.—S  =  ~ ~ 

r  —  1 

2.  The  first  term  is  4,  the  ratio  3,  and  the  last  term  is  972  ; 
what  is  the  sum  of  the  terms  ? 

3.  What  sum  can  be  paid  by  8  instalments  ;  the  first  being 
II,  the  second  $2,  etc.,  in  a  geometrical  series  ? 

4.  A  man  bought  a  dozen  sheep,  agreeing  to  pay  1  ct.  for 
the  first,  2  cts.  for  the  second,  4  cts.  for  the  third,  etc.;  what 
did  he  pay  for  the  12  sheep  ? 

5.  A  housekeeper  bought  12  chairs,  paying  2  cts.  for  the 
first,  6  cts.  for  the  second,  and  so  on  ;  what  did  they  cost  ? 

782.  To  find  the  Sum  of  a  Descending  Infinite  Series,  when  the 
First  Term  and  Ratio  are  given. 

Note. — In  a  descending  infinite  series  the  last  term  being  infinitely 
small,  is  regarded  as  0.     Hence,  the 

Rule. — Divide  the  first  term  by  the  difference  between 
the  ratio  and  1,  and  the  quotient  ivill  be  the  sum  required. 

1.  What  is  the  sum  of  the  series  f,  ^,  -J-.  ^,  continued  to 
infinity,  the  ratio  being  |^?     Ans.  1^. 

Note. — The  preceding  problems  in  the  Progressions  embrace  their  ordi- 
nary applications.  Others  might  be  given,  but  they  involve  principles 
with  which  the  pupil  is  not  supposed  to  be  acquainted. 

Questions. 

760.  Wliat  is  progression?  761.  The  terms?  763.  An  arithmetical 
progression  ?  765.  How  is  each  term  found  in  an  ascending  series  ?  766. 
In  a  descending  series  ? 

767.  Name  the  parts.  768.  How  find  the  last  term  ?  770.  The  first 
term?  771.  Number  of  terms.  773.  The  common  difference  ?  773.  The 
sum  of  all  the  terms  ? 

774.  What  is  geometrical  progression?  778.  Name  the  parts.  779. 
How  find  the  last  term  ?  780.  The  first?  781.  The  sum  of  all  the  terms? 
783.  The  sum  of  a  descending  infinite  series  ? 


ENSUFvATION. 


1^ 


V   ^ 


Definitions. 

783.  Mensuration  is  the  process  of  measuring  lines,  sur- 
faces, and  solids. 

784.  A  Line  is  length  without  breadth  or  thickness. 


785.  A  Straight  Line  is  one  that  does  not 
change  its  direction,  and  is  the  shortest  dis- 
tance between  two  points  in  the  same  plane. 


786.  Parallel  Lines  are  those  which  are 
equally  distant  from  each  other  at  every 
point. 

787.  Curved  Lines  are  those  which  change 
their  direction  at  every  point. 

788.  A  Horizontal  Line  is  one  that  is  parallel  to  the  horizon 
or  water  level. 

789.  A  Perpendicular  Line  is  a  straight  line  meeting  another 
straight  line,  so  as  to  make  the  two  adjacent  openings  equal. 
As  AB  and  CD.     (Art.  792.) 

790.  A  Perpendicular  to  a  horizontal  line  is  called  a  Vertical 
line. 


791.  A  Plane  Angle  is  the  opening  be- 
tween two  straight  lines  drawn  from  the 
same  point. 

Thus,  the  opening  between  AB  and  AC  is  an  an- 
gle, the  lines  AB  and  AC  are  called  the  sides,  and 
the  point  A  the  vertex  of  the  angle. 


*.A 

^:-\ 

'  s 

* 

t 

% 

« 

D 


320  Mensuration, 

792.  A  Right  Angle  is  one  of  the  two  equal 
angles  formed  by  the  meeting  of  two  straight 
lines  perpendicular  to  each  other. 

Thus,  the  adjacent  angles  ABC  and  ABD  are  liofht 
angles,  and  the  lines  AB  and  CD  are  perpendicular  to 
each  other. 

793.  An  Acute  Angle  is  one  that  is 
less  than  a  right  angle ;  as  AOB. 

A' 

794.  An  Obtuse  Angle  is  one  that  is  ^ 
greater  than  a  right  angle  ;  as  BCD. 

Note. — All  angles  except  right  angles  are  called  oblique  angles. 

795.  A  Surface  is  that  which  has  length  and  breadth,  with- 
out thickness. 

Surfaces  are  either  plane  or  curved.     The  surface  of  a  table  is  plane, 
that  of  an  orange  is  curved. 

796.  A  Plane  Figure  is  one  which  represents  a  surface  all 
the  parts  of  which  are  in  the  same  plane. 

797.  A  Polygon  is  a  plane  figure  bounded  by  three  or  more 
straight  lines. 

798.  The  Perimeter  of  a  polygon  is  the  line  by  which  it  is 
bounded. 

799.  A  Regular  Polygon  has  all  its  sides  and  all  its  angles 
equal. 

800.  A  polygon  having  three  sides  is  called  a  triangle  ;  four 
sides,  a  quadrilateral ;  five  sides,  ^  pentago7i ;  six  sides,  a  hex 
agon  ;  seven  sides,  a  heptago7i ;  eight  sides,  an  octagon  ;  etc. 

801.  A  Triangle  is  a  polygon  having  three  sides  ^ 
and  three  angles. 

802.  The  Base  of  a  triangle  is  the  side  AB  on 
which  it  is  supposed  to  stand. 

A      D     B 

803.  A  Vertical  Angle  is  the  angle  opposite  the  base;  as  C. 


Area  of  Triangles, 


321 


804.  The  Altitude  of  a  triangle  is  the  perpendicular  CD 
drawn  from  the  vertical  angle  to  the  base. 

805.  An  Equilateral  Triangle  is  one  having  three   equal 
sides. 


Isosceles. 


Scalene. 


Equilateral. 

806.  An  Isosceles  Triangle  is  one  having  only  two  equal 
sides. 

807.  A  Scalene  Triangle  is  one  having  all  its  sides  unequal. 


Area    of    Triangles. 

808.  It  is  proved  by  Geometry  that 

Tlie  area  of  a  triangle  is  equal  to  half  the  area  of  a  parallelo- 
jram  of  equal  base  and  altitude. 

Illustration. — Let  ABCD  be  a  paralle''ogram 
whose  altitude  is  the  perpendicular  J^B. 

Connect  the  diagonal   corners  by  the  straight 
line  BD,  and  the  parallelogram  will   be  divided 
into    two  equal  triangles,  the  altitude  of  each      a      e 
being  EB. 

The  area  of  a  parallelogram  or  rectangle  is  equal  to  the  length  multi- 
plied by  the  breadth . 

809.  To  find  the  Area  of  a  Triangle  when  the  Base  and  Altitude 

are  given. 

1.  What  is  the  area  of  a  triangle  whose  base  is  30  ft.  and  its 
altitude  12  feet  ? 

Let  the  base  AD  of  the  triangle  ABD  be  30  ft.,  and  EB,  its  altitude,  be 
12  ft. 

Then  30  x  12  =  360  sq.  ft.,  the  area  of  the  parallelogram. 

And  30  X  6  (1^  the  altitude)  =  180  sq.  ft.,  area  of  triangle.     Hence,  the 

Rule. — Multiply  the  base  hy  half  the  altitude. 


323  Mensuration, 

2.  What  is  tlie  area  of  a  triangle  wliose  base  is  45  feet,  and 
its  altitude  20  feet  ? 

3.  What  is  the  area  of  a  triangle  whose  base  is  156  feet,  and 
its  altitude  63  feet  ? 

4.  Find  the  number  of  acres  in  a  triangular  field  whose 
base  is  227  rods  and  altitude  65  rods. 

5.  What  is  the  area  of  a  triangle  whose  base  is  135  yds.,  and 
its  altitude  is  half  its  base  ? 

6.  Find  the  number  of  sq.  feet  in  the  gable  end  of  a  build- 
ing 40  ft.  wide,  and  12 J^  ft.  from  the  beam  to  the  ridgepole. 

810.  To  find  the  Area   of  a   Triangle,   when    the    Three  SIcfes 

are  given. 

From  half  the  sum  of  the  three  sides  subtract  each  side  respec- 
tively ;  then  multiply  half  the  sum  and  the  three  remainders 
together,  and  extract  the  square  root  of  the  product. 

1.  What  is  the  area  of  a  triangle  whose  sides  are  respectively 
10  feet,  12  feet,  and  16  feet  ? 

Solution.— (10  +  12 +  l6)-=-2  =  19  feet. 
19-10  =  9  ;  19-12  =  7  ;  19  -16  =  3. 


Now  19x9x7x3  =  3591,  and  y'3591  =  59.92+  sq.  ft. 

2.  What  is  the  area  of  an  equilateral  triangle  whose  side  is 
12  yds.  ? 

3.  What  is  the  area  of  an  isosceles  triangle  whose  base  is 
30  feet  and  sides  20  feet  ? 

4.  How  many  acres  in  a  triangular  field  whose  sides  are  45, 
53,  and  64  rods  ? 

811.  To  find  the  Altitude,  when  the  Area  and  Base  are  given. 
Rule. — Divide  the  area  by  half  the  base. 

1.  What  is  the  altitude  of  a  triangle  whose  area  is  27-1  square 
yards  and  base  5  yards  ?     Ans.  11  yards. 


Quadrilaterals. 


323 


2.  What  is  tlie  altitude  of  a  triangle  whose  area  is  210  sq. 
yds.  and  its  base  140  yards  ? 

3.  What  is  the  altitude  of  a  triangle  whose  base  is  150  rods 
and  its  area  11250  square  rods  ? 

812.  To  find  the  Base,  when  the  Area  and  Altitude  are  given. 
KuLE. — Divide  the  area  hy  half  the  altitude. 

1.  What  is  the  base  of  a  triangle  whose  area  is  154  sq.  ft. 
and  its  altitude  14  feet?     Arts.  22  feet. 

2.  What  is  the  base  of  a  triangle  whose  area  is  40  acres  and 
its  altitude  160  rods? 

3.  Find  the  base  of  a  triangle  whose  area  is  5260  sq.  yd.,  and 
altitude  200  yards. 


Quadrilaterals. 

813.  A  Quadrilateral  is  a  polygon  bounded  by  four  straight 
lines. 

A  quadrilateral  is  either  a  mTallelogram,  a  trapezoid,  or  a  trapezium. 


814.  A  Parallelogram  is  a  quadrilateral 
haying  its  opposite  sides  equal  and  parallel. 

815.  The  Altitude  of  a  quadrilateral  hav- 
ing two  parallel  sides  is  the  perpendicular 
distance  between  these  sides  ;  as,  AL. 

816.  A  Rectangle  is  a  right-angled  parallel- 
ogram. 


Note. — When  the  four  sides  of  a  rectangle  are  equal 
it  is  called  a  square.     (Art.  345.) 


817.  A  Rhomboid  is  an  oblique-angled  par- 
ielogram. 

818.  A  Rhombus  is  an  equilateral  rhomboid. 


324 


Mensuration. 


819.  A  Trapezoid  is  a  quadrilateral 
which  has  two  of  its  sides  parallel. 

820.  A  Trapezium  is  a  quadrilateral 
having  four  unequal  sides,  no  two  of 
which  are  parallel.* 

Note. — The  Diagonal  of  a  plane  figure  is      a 
a  straight  line  connecting  two  of  its  angles 
not  adjacent ;  as  AB. 


821.  To  find  the  Area  of  a   Parallelogram,  when   the   Base   and 

Altitude  are  given. 

1.  What  is  the  area  of  a  rectangle  whose  base  is  88  feet  and 
altitude  30  feet  ? 

Solution.— 58  x  30  =  1740  sq.  ft.,  Ans. 

2.  What  is  the  area  of  a  rhomboid  whose  base  is  63  feet  and 
its  altitude  40  feet  ? 

Solution.— 63  x  40  =  2520  sq.  ft.,  Ans.    Hence,  the 

EuLE. — Multiply  the  base  by  the  altitude. 

Note. — The  area  of  a  square,  a  rectangle,  a  rhomboid  and  rhombus  is 
found  in  the  same  manner. 

3.  How  many  acres  in  a  field  120  rods  long,  and  90  rods 
wide  ? 

4.  How  many  acres  in  a  field  800  rods  long,  and  128  rods 
wide  ? 

5.  Find  the  area  of  a  square  field  whose  sides  are  65  rods  in 
length. 

6.  A  man  fenced  off  a  rectangular  field  containing  3750  sq. 
rods,  the  length  of  which  was  75  rods  ;  what  was  its  breadth  ? 

7.  One  side  of  a  rectangular  field  is  1  mile  in  length,  and  it 
contains  160  acres  ;  what  is  the  length  of  the  other  side? 

*  The  majority  of  Authors  define  these  terms  as  in  the  text.  Others,  among  whom 
arc  Le2;endre,  Dr.  Brewster,  Younjr,  and  De  Morgan,  apply  the  definition  here  given  of 
A  Trapezium  to  the  Trapezoid,  and  vice  versa. 


Quadrilaterals,  325 

8.  The  length  of  a  rhombus  is  17  ft.,  and  its  perpendicular 
height  16  ft.;  what  is  its  area?  Ans.  272  sq.  ft. 

9.  What  is  the  area  of  a  rhomboid  whose  altitude  is  25  rods, 
and  its  length  28.6  rods  ? 

822.  To  find  the  Area  of  a  Trapezoid,  w^en  its  Parallel  Sides 

and  Altitude  are  given. 

1.  Find  the  area  of  a  trapezoid  whose  parallel  sides  are  28 
and  36  feet  and  its  altitude  12  feet. 

Solution.— The  sum  of  the  parallel  sides  28  +  36  =  64  ft.,  i  of  64  =  32 
ft.,  and  32  ft.  x  13  (the  altitude)  =  384  sq.  ft.,  Ans.     Hence,  the 

EuLE. — Multiply  half  the  sum  of  the  parallel  sides  hy 
the  altitude. 

2.  The  parallel  sides  of  a  trapezoid  are  25  yd.  and  21  yd.,  and 
its  altitude  16  yd. ;  what  is  its  area  ? 

3.  Find  the  area  of  a  trapezoid  whose  parallel  sides  are  25 
rods  and  37  rods,  and  its  altitude  18  rods.  _        , 

823.  To  find  the  Area  of  a   Trapezhon,  when  the  Diagonal  and 

Perpendiculars  are  given. 

1.  A  man  bought  a  city  lot  in  the  form  >/r^\ 

of  a  trapezium,  the  diagonal  of  which  was     y^    \     ^\^^ 

84  ft.  and  perpendiculars  from  the  opposite  <^^ ^ -^ 

angles  12  ft.  and  16  ft. ;  what  was  its  area?      N.       |      ^^ 

Solution. — The  sum  of  the  perpendiculars  is  ^^<y^ 

28  ft. ;  1  of  28  -  14  and  84  ft.  x  14  =  1176  sq.  ft.,  Ans.    Hence,  the 

EuLE. — Multiply  the  diagonal  hy  half  the  sum  of  the 
perpendieulars  to  it  from  the  opposite  angles. 

2.  A  man  bought  a  meadow  in  the  form  of  a  trapezium,  the 
diagonal  of  which  was  250  rods,  and  the  perpendiculars  30  and 
35  rods;  how  many  acres  did  it  contain  ? 

3.  Find  the  area  of  an  irregular  piece  of  land,  the  diagonal 
of  which  is  320  yards,  and  the  perpendiculars  35.5  yards  and 
42J  yards. 


326 


Mensuration, 


Circles. 

824.  A  Circle  is  a  plane  figure  bounded 
by  a  curve  line,  every  part  of  wbicli  is 
equally  distant  from  a  point  within  called 
the  center. 

825.  The  Circumference  of  a  circle  is 
the  curve  line  by  Avhich  it  is  bounded. 

826.  The  Diameter  is  a  straight   line  drawn  through  the 
center,  terminating  at  each  end  in  the  circumference. 

827.  The  Radius  is  a  straight  line  drawn  from  the  center  to 
the  circumference,  and  is  equal  to  lialf  the  diameter. 

Note. — From  the  definition  of  a  circle,  it  follows  that  all  the  radii  are 
equal ;  also,  that  all  the  diameters  are  equal, 

828.  From  the  relation  of  the  circumference  and  diameter 
to  each  other,  we  derive  from  Geometry  the  following 

Principles. 

i°.   Tiie  Circumference  =  the  Diameter  x  S.I4.I6  nearly. 

2°.   The  Diameter  of  a  Circle  =  the  Circumference  -^  3.  IJflG 
nearly. 

829.  To  find  the  Circumference  of  a  Circle,  when  the  Diameter 

is  given. 

1.  What  isifcthe  circumference  of  a  circle  whose  diameter  is 
15  feet? 

Solution.— 15  ft.  x  3.1416  =  47.125  ft.,  Ans.    Hence,  the 

EuLE.  —  Multiply    the  given    diameter    by    3.1416. 
(Art.  828,,  i°.) 

2.  What  is  the  circumference  of  a  circle  whose  diameter  is 
45  yards  ? 

3.  What  is  the  circumference  of  a  circle  whose  diameter  is 
100  rods  ? 


Circles.  327 

830.  To  find  the  Diameter  of  a  Circle,  when  the  Circumference 

is  given. 

1.  What  is  the  diameter  of  a  circle  whose  circumference  is 
65^  feet  ? 

Solution.— 65.5-5-3.1416  =20.849+  ft.,  Am.    Hence,  the 

Rule. — Divide  the  circinnference  by  S.I4I6.    (Prin.5°.) 

2.  What  is  the  diameter  of  a  circle  whose  circumference  is 
94.2477  rods  ? 

3.  What  is  the  diameter  of  a  circle  whose  circumference  is 
628.318  yards? 

Note. — The  diameter  of  a  circle  may  also  be  found  by  dividing  the 
area  by  .7854,  and  extracting  the  square  root  of  the  quotient. 

4.  Required   the   diameter   of  a  circle   containing  50.2656 
square  rods. 

5.  Required  the  diameter  of  a  circle  containing  201.0624 
square  feet. 

831.  To  find  the  Area  of  a  Circle,  when  the  Diameter  and  Cir- 

cumference are  given. 

1.  What  is  the  area  of  a  circle  whose  diameter  is  10  ft.  and 
circumference  31.416  ft.? 

Solution.— 3J-.f  16  x^^  =  78.54  sq.  ft.,  Ans. 

Or,  31.416  X  (10^4)  =  78.54  sq.  ft.,  Ans.     Hence,  the 

Rule. — Multiply  half  the  circuinference  by  half  the 
diameter;  or, 
Multiply  the  circumference  by  a  fourth  of  the  diameter. 

Notes. — 1.  If  only  one  of  these  dimensions  are  given,  the  other  must 
be  found  before  the  rule  can  be  applied.     (Ex.  3,  4.) 

2.  The  area  of  a  circle  may  also  be  found  by  multiplying  the  square  of 
its  diameter  by  the  decimal  .7854. 

2.  Find  the  area  of  a  circle  whose  diameter  is  20  ft.  ? 
Solution.— 202  ^  .7354  =  314.16  sq.  ft.,  Ans. 

3.  What  is  the  area  of  a  circle  whose  diameter  is  100  ft.  ? 

4.  AYhat  is  the  area  of  a  circle  whose  diameter  is  120  rods  ? 


328 


Mensu7'ation, 


5.  What   is   the   area  of  a  circle  whose   circumference  is 
160  yards  ? 

6.  What  is  the  diameter  of  a  wheel  whose  circumference  is 

50  ft.  ? 

7.  Find    the    circumference    of   a    tree  whose    diameter  is 
3  ft.  4  in. 

8.  What  is  the  area  of  a  circle  whose  radius  is  15  ft.  ? 

9.  How  many  acres  in  a  circular  park  whose  circumference 

is  2  miles  ? 

10.  What  is  the  radius  of  a  circle  which  contains  IJ  acre  ? 

Solids. 


832-  A    Solid    is    that    which    has    length,    breadth,    and 
thickness. 

833.  A  Prism  is  a  solid  whose  bases  are  similar,  equal,  and 
parallel,  and  whose  sides  are  parallelograms. 

Note. — Prisms  are  named  from  the  form  of  tlieir  bases,  as  trinnguJar, 
quadrangular,  pentagonal,  hexagonal,  etc. 

834.  A  Right  Prism   is  one 

whose  sides  are  perpendicular  to 
its  bases. 

835.  A    Triangular  Prism  is 

one  whose  bases  are  triangles. 

836.  A  Rectangular  Prism  is 

one  whose  bases  are  rectangles,  and  its  sides  perpendicular  to 
its  bases. 

837.  The  Lateral  Surface  of  a  prism  is  the  sum  of  all  its 
faces. 

838.  The  Altitude  of  a  prism  is  the  perpendicular  distance 
between  its  bases. 

839.  All  rectangular  solids  are  prisms. 


Solids.  329 

Notes. — 1.  When  their  sides  are  all  equal  to  each  other  they  are 
called  cubes. 

3.  When  their  bases  are  parallelograms  they  are  called  paraUelopipeds, 
or  parallelopipedons. 

840.  A  Cylinder  is  a  circular  body  of  i 
uniform  diameter,  whose  ends  are  equal  \ 
parallel  circles. 

841.  To  find  the  Lateral  Surface  of  a  Prism  or  Cylinder. 

1.  What  is  the  lateral  surface  of  a  prism  whose  altitude  is 
12  ft. ,  and  its  base  a  pentagon  each  side  of  which  is  6  feet  ? 

Solution. — 6  ft.  x  o  =  30  ft.  the  perimeter. 
30  ft.  X  12  =  360  sq.  ft.  surface,  Ans. 

2.  What  is  the  convex  surface  of  a  cylinder  32  inches  in  cir- 
cumference and  40  inches  long? 

Solution. — 33  x  40  =  1280  sq,  in.,  Ans.    Hence,  the 

EuLE. — Multiply  the  perimeter  of  the  base  hy  the 
altitude. 

Note. — To  find  the  entire  surface,  the  area  of  the  bases  must  be  added 
to  the  lateral  surface. 

3.  What  is  the  surface  of  a  triangular  prism  whose  altitude 
is  91  feet,  and  the  sides  of  its  base  are  3,  4,  and  5  ft.  respec- 
tively ? 

4.  Required  the  lateral  surface  of  a  triangular  prism  whose 
perimeter  is  44-  inches,  and  its  length  12  inches. 

5.  Required  the  lateral  surface  of  a  quadrangular  prism 
whose  sides  are  each  2  feet,  and  its  length  19  feet. 

6.  Required  the  convex  surface  of  a  log  whose  circumference 
is  18  ft.,  and  length  32  ft.? 

7.  What  is  the  convex  surface  of  a  cylinder  16  feet  in  cir- 
cumference and  40  feet  long  ? 

8.  What  is  the  convex  surface  of  a  cylinder  whose  diameter 
is  20  feet  and  its  height  65  feet  ? 


330 


Mensuration. 


842.  To  find  the   Contents  of  a   Prism   or  Cylinder,  when   the 
Perimeter  of  the  Base  and  the  Altitude  are  given. 

1.  What  are  the  contents  of  a  triangular  prism  whose  alti- 
tude is  10  ft.  and  perimeter  of  its  equilateral  base  36  feet  ? 

Solution.— 122  —  6'  =  108. 

-V/IOS  =  10.4  nearly,  altitude  of  base.     (Art.  744.) 
Again,  12  x  5.2  =  62.4  sq.  ft.,  area  of  base.     (Art.  809.) 
62.4  sq.  ft.  X  10  =  624  cu.  ft.,  contents. 

2.  What  are  the  contents  of  a  cylinder  whose   altitude   is 
6  ft.  6  in.  and  the  diameter  of  its  base  3  ft.  ? 

Solution.— 32  x  .7854  =  7.0686  sq.  ft.,  area  of  base.    (Art.  831,  n.) 
7.0686  sq.  ft.  x  6.5  =  45.9459  cu.  ft.,  contents.     Hence,  the 

EuLE. — Multiply  the  area  of  the  base  by  the  altitude. 

Note. — This  rule  is  applicable  to  all  prisms,  triangular,  quadrangular, 
etc, ;  also  to  all  parallelopipedons. 

3.  What  is  the  solidity  of  a  prism  whose  base  is  5  feet  square, 
and  its  height  15  feet  ? 

4.  AVhat  is  the  solidity  of  a  triangular  prism  whose  height 
is  20  feet,  and  the  area  of  whose  base  is  460  square  feet  ? 

5.  Eequired  the  solidity  of  a  cylinder  6  feet  in  diameter,  and 
20  feet  high. 

6.  Eequired  the  solidity  of  a  cylinder  30  feet  in  diameter, 
and  65  feet  long. 


Pyramid. 


Frustum. 


Cone. 


iYustum. 


843.  A  Pyramid  is  a  solid  whose  base  is  a  ti^iangle,  sqnare, 
or  polygon,  and  whose  sides  terminate  in  a  point,  called  the 
vertex. 


Note. — The  sides  which  meet  in  the  vertex  are  triangles. 


Solids. 


331 


844.  A  Cone  is  a  solid  which  has  a  circle  for  its  base,  and 
terminates  in  a  point  called  the  vertex. 

845.  A  Frustum  of  a  pyramid  or  cone  is  the  part  which  is 
left  after  the  top  is  cut  off  by  a  plane  parallel  to  the  base. 

846.  To  find  the   Contents  of  a   Pyramid  or  a  Cone,  when  the 

Base  and  Altitude  are  given. 

1.  What  are  the  contents  of  a  pyramid  whose  base  is  144  sq. 
feet,  and  its  altitude  30  feet  ? 

Solution.— 144  sq.  ft.  x  10  (i  of  altitude)  =  1440  cu.  ft.,  Ans. 

2.  What  are  the  contents  of  a  cone  the  area  of  whose  base  is 
1864  sq.  feet,  and  its  altitude  36  feet  ? 

Solution.— 1864  x  12  (i  of  altitude)  =  22368  cu.  ft.     Hence,  the 

Rule. — Multiply  the  ctvea  of  the  base  hy  ^  of  the 
ciltitiide. 

Note. — The  contents  of  a  frustum  of  a  pyramid  or  cone  are  found  hy 
adding  the  areas  of  the  two  ends  to  the  square  root  of  the  product  of  those 
areas,  and  multiplying  the  sum  by  ^  of  the  altitude. 

3.  Wliat  are  the  contents  of  a  pyramid  whose  base  is  22  ft. 
square,  and  its  altitude  48  ft.? 

4.  Of  a  cone  45  ft.  high,  whose  base  is  18  ft.  diameter  ? 

5.  The  altitude  of  a  frustum  of  a  pyramid  is  27  ft.,  the  ends 
are   5   ft.   and   3   ft.   square ;    what   is  its 
solidity  ? 


847.  A  Sphere  or  Globe  is  a  solid  ter- 
minated by  a  curve  surface,  every  part  of 
which  is  equally  dista^it  from  a  point  with- 
in, called  the  center. 


848.  The  Diameter  of  a  sphere  is  a  straight  line  drawn 
tbrough  its  center  and  terminated  at  both  ends  by  the  surface. 

849.  The  Radius  of  a  sphere  is  a  straight  line  drawn  from 
its  center  to  any  point  in  its  surface. 


332  Mensuration, 

850.  To  find  the  Surface  of  a  Sphere,  the  Circumference  and 

Diameter  being  given. 

1.  Required  the  surface  of  a  globe  8  inches  in  diameter. 

Solution.— 8  x  3.1416  =  25.1328  in. 
25.1328  X  8  =  201.0624  sq.  in.    Hence,  the 

Rule. — Multiply  the  circumference  hy  the  diameter, 

2.  Required  the  surface  of  a  15  inch  globe.   Ans.  4.91  sq.  ft. 

3.  Required  the  surface  of  the  earthy  its  diameter  being 
8000  miles. 

851.  To   find   the   Solidity   of  a   Sjihere,  the   Surface   and 

)  Diameter  being  given. 

1.  Find  the  solidity  of  a  sphere  whose  diameter  is  15  inches 
and  its  surface  4.91  sq.  feet  ? 

Solution.— 4.91  x  144  =  707.04  sq.  in. 

707.04  sq.  in.  x  2.5  =  1767.6  cu.  in.,  A71S.     Hence,  the 

Rule. — Multiply  the  surface  hy  ^  of  the  diameter, 

2.  What  is  the  solidity  of  a  10-inch  globe  ? 
Ans.  523.6  cu.  in. 

3.  What  is  the  solidity  of  the  earth,  its  surface  being 
196900278  sq.  miles,  and  its  mean  diameter  7916  miles? 

4.  Find  the  solidity  of  a  cannon  ball  9  inches  in  diameter  ? 

852.  To  measure  the  height  of  an  object  standing  in  a  plane. 

1.  What  is  the  height  of  a  tree  standing  in  a  plane  which 
casts  a  shadow  50  feet,  measured  with  a  pole  6  ft.  long,  casting 
a  shadow  10  ft.  ? 

Solution. — Take  a  pole  of  any  convenient  length,  and  placing  it  in  a 
perpendicular  position,  measure  the  length  of  its  shadow,  which  we  will 
suppose  to  be  10  feet,  then  say 

10  ft.  (shadow  of  p.)  :  50  ft.  (shadow  of  t.)  : :  6  ft.  (1.  of  p.)  :  height  of  tree. 

50  X  6  =  300,  and  300-^-10  =  30  feet. 

2.  What  is  the  height  of  a  pyramid  standing  in  a  plane  which 
casts  a  shadow  of  100  feet,  measured  with  a  pole  8  ft.  long 
which  casts  a  shadoAV  of  12  feet? 


Gauging  of  Casks,  333 


Gauging    of    Casks. 

853.  Gauging  is  finding  the  capacity  or  contents  cf  casks 
and  otlier  vessels. 

854.  The  mean  diameter  of  a  cask  is  equal  to  half  the  snm 
of  the  head  diameter  and  bung  diameter.     (Art.  766,  X.  2.) 

Note. — The  contents  of  a  cask  are  equal  to  those  of  a  cylinder  having 
the  same  length  and  a  diameter  equal  to  the  mean  diameter  of  the  cask. 

855.  To  find  the   Contents  of  a   CasJ^-,  when   its  Length,   Its 

Head,  and  Bung  Diameters  are  given. 

1.  How  many  gallons  in  a  cask  whose  length  is  35  inches,  its 
bung  diameter  30  inches,  and  head  diameter  20  inches? 

Solution.— (30  + 26)-^2  =  28  in.,  the  mean  diameter.    (Art.  854.) 

282  X  .7854  =  area  of  base. 

Area  of  base  x  length  =  contents  in  cubic  inches,  which  is  reduced  to 
gallons  by  dividing  by  231. 

Instead  of  using  the  factor  .7854,  if  we  divide  it  by  231,  the  number  of 
cubic  inches  in  a  gallon,  and  multiply  by  the  quotient  .0034,  the  operation 
is  shortened,  and  the  result  is  in  gallons.     Thus. 

282  X  35  X  .0034  =  93.296  gal.,  Ans.    Hence,  the 

Rule. — Multiply  the  square  of  the  mean  diameter  by 
the  length  in  inches,  and  this  product  by  .0034  fo^^ 
gallons,  or  by  .0129  for  liters. 

Note. — In  fiuding  the  contents  of  cisterns,  it  is  suflBciently  accurate  for 
ordinary  purposes  to  call  a  cubic  foot  =  7|  gallons. 

2.  What  is  the  capacity  of  a  cask  whose  length  is  30  inches, 
its  head  diameter  18,  and  bung  diameter  24  inches  ? 

3.  Find  the  contents  in  liters  of  a  cask  whose  length  is 
54  inches,  its  bung  diameter  42,  and  head  diameter  36  inches? 

4.  Find  the  contents  in  gallons  of  a  rectangular  cistern 
44  ft.  long,  3i  ft.  wide,  and  5  ft.  deep. 


334  Mensuration, 


Tonnage    of    Vessels. 

856.  Tonnage  is  the  weight  in  tons  which  a  vessel  will  carry. 
It  is  estimated  by  the  following 

Carpenter's     Rule. 

Multiply  together  the  length  of  the  heel,  the  breadth  at 
the  jnaifi  beam,  and  the  depth  of  the  hold  in  feet,  and 
divide  the  product  by  95  (the  cu.  ft,  allowed  for  a  ton) ; 
the  result  will  be  the  tonnage. 

For  a  double  decker,  instead  of  the  depth  of  the  hold, 
taJce  half  the  breadth  of  the  beam. 

Note. — A  RegiHer  Ton  =  100  cu.  ft.  is  the  legal  standard. 

A  Shipping  Ton  =  ]  ^^      '  ft*'  F "      '       \  ^^^^  ^^  estimating  cargoes. 

1.  What  is  the  tonnage  of  a  double  decker  with  300  ft.  keel 
and  40  ft.  beam  ?    Ans.  25261^9  tons. 

2.  AVhat  is  the  tonnage  of  a  single  decked  vessel  whose 
length  is  100  ft.,  the  breadth  30  ft.,  and  the  depth  12  ft.  ? 

Questions. 

783.  What  is  mensuration  ?  784.  A  line  ?  785.  Straight  line  ?  786. 
Parallel  lines ?  789.  Perpendicular  line ?  791.  What  is  a  plane  angle? 
The  vertex?    792.  A  right  angle ?    793.  Acute?    794  Obtuse? 

795.  What  is  a  surface ?  796.  A  plane  figure?  797.  A  polygon?  801. 
A  triangle  ?  803.  Vertical  angle  ?  804.  The  altitude  of  a  triangle  ?  805. 
An  equilateral  triangle ?    806.  Isosceles?    807.  Scalene? 

809.  How  find  the  area  of  a  triangle  from  the  base  and  altitude  ?  810. 
How  when  the  three  sides  are  given  ?  813.  What  is  a  quadrilateral  ? 
Name  the  three  kinds  of  quadrilaterals.  814.  A  parallelogram  ?  815.  Al- 
titude of  a  quadrilateral?  816.  A  rectangle?  817.  A  rhomboid?  818. 
Rhombus  ?  821.  How  find  the  area  of  a  parallelogram  ?  822.  Of  a  trape- 
zoid ?    828.  Of  a  trapezium  ? 

824.  What  is  a  circle?  825.  The  circumference?  826.  Diameter? 
827.  Radius  ?  829.  How  find  the  circumference  when  diameter  is  given  ? 
830.  How  find  diameter  when  circumference  is  given  ? 

832.  What  is  a  solid  ?  833.  A  prism  ?  834.  A  right  prism  ?  835.  Tri- 
angular? 836.  Rectangular?  840.  What  is  a  cylinder?  841.  How  find 
the  lateral  surface  of  a  prism  or  cylinder  ?  842.  When  the  perimeter  of 
the  base  and  the  altitude  are  given,  how  find  the  contents  ? 

847.  What  is  a  sphere?    850.  How  find  the  surface? 


QUESTIONS. 

1^-- — s — ^ — ^  •' — 

I.    Oral     Exercises. 

857.  1.  A  lad  earned  $f  the  first  day,  ||  the  second  dajj 
and  in  both  days  he  spent  |-|-;  how  much  had  he  left  ? 

2.  A  frog  at  the  bottom  of  a  well  jumped  up  SJ  yards  the 
first  day,  and  2  yd.  the  second,  but  afterwards  fell  back  1^  yd.; 
how  far  was  he  then  from  the  bottom  of  the  well? 

3.  From  a  bin  containing  10|^  bushels  of  wheat,  a  miller  took 
S-^Q  bushels  at  one  time,  and  2|-  bushels  at  another ;  how  much 
w^heat  remained  in  the  bin  ? 

4.  If  you  buy  a  melon  for  18f  cents,  and  a  quart  of  black- 
berries for  12|^  cents,  and  pay  the  market-man  50  cents,  how 
much  change  ought  he  to  give  you  ? 

5.  What  number  must  be  taken  from  12,  that  the  remainder 
may  be  5  J  ? 

6.  What  number  added  to  itself  three  times  will  make  48? 

7.  What  number  added  to  ^  of  itself  will  make  36  ? 

8.  A  boy  counting  his  money  said,  if  he  had  18f  cents  more, 
he  would  then  have  62^  cents  ;  how  much  had  he? 

9.  What  is  the  sum  of  37|  and  6^  ?     What  the  difference  ? 

10.  A  man  having  865,  paid  I24|-  for  a  cow,  and  $15|  for  a 
ton  of  hay  ;  how  much  money  had  he  left  ? 

11.  A  man  having  3J  acres  of  land,  bought  4^  acres  more ; 
afterwards  he  sold  2i  acres.     How  much  land  had  he  then  ? 

12.  A  farmer  paid  $3^  apiece  for  sheep ;  how  many  did  he 
bay  for  111 50  ? 

13.  At  $12|^  an  acre,  how  many  acres  can  be  bought  for  $500  ? 

14.  If  a  man  earns  $33J  a  month,  how  long  will  it  take  him 
to  earn  $200  ? 

15.  75  is  f  of  what  number  ?     J  of  what  number  ? 

16.  Henry  lost  35  yards  of  his  kite  line,  which  was  f  of  its 
whole  length  ;  how  long  was  it  ? 


336  Ted  Questions, 

17.  A  man  sold  a  watch  for  $60,  which  was  |  the  cost ;  what 
did  it  cost  ? 

18.  If  7  barrels  of  peanuts  cost  $42,  what  will  11  barrels 
cost? 

19.  Bought  a  bag  of  coffee,  weighing  60  lbs.,  for  $20 ;  what 
must  I  sell  it  for  to  make  12^  profit  ? 

20.  Bought  a  horse  and  buggy  for  $350 ;  the  price  of  the 
buggy  was  f  as  much  as  the  horse.  What  was  the  price  of  the 
horse  ? 

21.  Divide  48  into  three  such  parts,  that  the  first  shall  be 
twice  the  secoud,  and  the  third  3  times  the  second. 

22.  In  48256  metres,  how  many  kilometers  ? 

23.  If  the  Slim  of  two  numbers  is  34,  and  their  difference  8, 
what  are  the  numbers? 

24.  The  quantity  of  land  in  two  pastures  is  45  acres,  and  the 
difference  in  their  size  is  9  acres ;  how  many  acres  does  each 
contain  ? 

25.  If  the  difference  between  two  numbers  is  9,  and  their 
sum  32,  what  are  the  numbers  ? 

26.  If  the  difference  of  two  numbers  is  7^,  and  their  sum  22|-, 
what  are  the  numbers  ? 

27.  The  sum  of  the  ages  of  two  persons  is  87|-  years,  and  the 
difference  12|-  years;  what  are  their  ages? 

28.  What  number  is  that,  ^  and  \  of  which  is  equal  to  f 
of  35? 

29.  The  number  of  scholars  in  a  certain  school  is  75,  and  the 
boys  exceed  the  girls  by  13  :  how  many  of  each  sex  does  the 
school  contain  ? 

30.  A  third  of  the  trees  in  a  certain  orchard  are  pear  trees, 
\  are  peach  trees,  and  the  rest,  being  21,  are  plum  trees;  how 
mauy  trees  are  there  in  the  orchard? 

31.  A  man  paid  $150  for  a  horse,  and  sold  it  at  20  per  cent 
advance  ;  how  much  did  he  make  by  the  operation? 

32.  What  is  33i  per  cent  of  600  ?  Of  660?  1500?  2100? 
2700?     3600? 

33.  If  yon  have  $400,  and  lose  40  per  cent  of  it,  how  many 
dollars  will  you  lose  ? 


Test  Questions.  337 

34.  A  lad  having  500  marbles,  lost  50  per  cent  of  them ;  how 
many  did  he  lose  ?     How  many  did  he  have  left  ? 

35.  If  you  have  200  acres  of  land,  and  sell  75  per  cent  of  it, 
how  many  acres  will  you  sell  ? 

36.  A  man  having  -151000,  invested  it  in  a  speculation  by 
which  he  lost  100%'  of  it;  how  much  had  he  left? 

37.  A  lad  having  20  oranges,  gave  away  12  of  them ;  what 
per  cent  did  he  give  ? 

38.  There  are  6  working  days  to  1  sabbath ;  what  is  the  per- 
centage of  sabbaths  to  days  for  labor  ? 

39.  In  a  certain  state  prison,  2  out  of  3  of  the  inmates  are 
intemperate ;  what  is  the  percentage  of  intemperate  ? 

40.  What  per  cent  of  a  number  is  J  of  that  number  ? 

41.  A  farmer  bought  a  cart  and  a  plough  for  881 ;  the  price 
of  the  cart  was  8  times  that  of  the  plough.  What  was  the 
price  of  each  ? 

42.  An  agent  sold  a  horse  for  $200,  and  received  12|-  per 
cent  commission ;  how  much  did  he  receive  ? 

43.  What  part  of  one  year  is  6  months?  4  mo.  ?  3  mo.  ? 
2  mo.  ?     1|  mo.  ?     1^  mo.  ?     1  mo.  ? 

44.  What  part  of  a  year  is  8  months  ?     9  mo.  ?     10  mo.  ? 

45.  What  is  the  int.  of  1120  for  2  mo.,  at  4  per  cent  ? 

46.  What  is  the  int.  of  1250  for  6  mo.,  at  3  per  cent  ? 

47.  What  is  the  expense  of  collecting  a  tax  of  $500,  at  6  per 
cent  commission  ? 

48.  What  is  the  commission  on  1600,  at  12^  per  cent  ? 

49.  AVhat  is  the  interest  of  $200  for  1  year  and  3  months, 
4t  6  per  cent  ? 

50.  What  is  the  interest  of  $50  for  4  years,  at  G  per  cent  ? 

51.  What  is  the  amount  of  $100  for  3  years,  at  7  per  cent? 

52.  What  is  the  amount  of  $200  for  5  years,  at  6  per  cent? 

53.  The  product  of  A,  B,  and  C's  ages  is  240  years ;  A  is 
6  years,  and  B  is  5  years  old.     How  old  is  C  ? 

54.  Thomas  bought  36  apples  for  25  cents,  and  sold  them  at 
the  rate  of  4  for  3  cents ;  how  much  did  he  make  or  lose  ? 

55.  If  he  had  sold  them  at  the  rate  of  3  for  2  cents,  what 
would  have  been  the  result  of  his  bargain  ? 


338  Test  Questions, 

56.  If  a  market-man  buys  oranges  at  the  rate  of  3  cents 
apiece,  and  sells  2  for  7  cents,  what  per  cent  is  his  profit  ? 

57.  A  man  sold  a  mirror  for  132,  and  thereby  made  33|^  per 
cent;  what  per  cent  would  he  have  made  had  he  sold  it  for  $48  ? 

58.  A  and  B  hired  a  pasture  for  $36 ;  A  put  in  4  horses  for 
12  weeks  and  B  6  horses  for  10  weeks.  How  much  ought  each 
to  pay  ? 

59.  If  -J-  of  a  pole  stands  in  the  mud,  f  of  it  in  water,  and 
12  feet  are  above  w^ater,  what  is  the  length  of  the  pole,  and  how 
many  feet  in  each  part  ? 

60o  If  a  herring  and  a  half  cost  a  penny  and  a  half,  how 
many  can  you  buy  for  11  pence  ? 

61.  A  can  dig  a  cellar  in  3  weeks,  B  in  4  weeks,  and  C  in 
6  weeks;  how  long  will  it  take  all  three  to  dig  it  ? 

62.  A  farmer  having  rye  and  wheat  worth  6s.  and  10s.  a 
bushel,  wished  to  make  a  mixture  worth  9s.;  what  proportion 
of  each  must  he  put  in  ? 

63.  What  sum  at  1%  will  gain  184  int.  in  1  year  ? 

64.  A  grocer  having  two  kinds  of  tea,  worth  5s.  and  7s.  a 
pound,  mixed  5  pounds  of  each,  and  sold  the  mixture  at  6s.  6d. 
a  pound  ;  how  much  did  he  make  by  the  operation? 

65.  If  a  cistern  has  one  pipe  which  will  fill  it  in  8  hours,  and 
another  wiiich  will  empty  it  in  12  hours,  how  long  will  it  take 
to  fill  it,  if  both  run  together  ? 

66.  What  sum  at  %%  will  gain  $54  interest  in  1 J  year  ? 

67.  A  vat  holding  56  barrels  has  two  faucets,  one  of  which 
supplies  17  barrels  an  hour,  and  the  other  discharges  22  barrels 
an  hour ;  when  full,  how  long  will  it  take  to  empty  it,  when 
both  are  running? 

68.  What  is  the  difference  between  a  dozen  rods  square  and 
a  dozen  square  rods  ? 

69.  The  surface  of  a  cube  is  150  square  inches;  what  is  the 
length  of  its  edge  ? 

70.  Four  boats  start  at  the  same  time  from  Castle  Garden  to 
sail  round  Governor's  Island  ;  one  of  them  can  perform  the  trip 
in  2  hours,  another  in  3  hours,  another  in  4  hours,  and  the 
other  in  6  hours  ;  how  long,  if  they  continue  to  sail,  before  all 
will  meet  at  the  starting  place  ? 


Test  Questions.  339 

II.    Written     Exercjses, 

858.  1.  A  teacher  being  asked  how  many  pupils  he  had, 
replied  that  he  had  140  boys,  and  if  the  number  of  girls  were 
multiplied  by  the  number  of  boys,  the  product  would  be  22960  ; 
how  many  girls  had  he  ?     How  many  pupils  ? 

2.  The  length  of  a  rectangular  meadow  is  842  rods,  and  the 
product  of  the  length  and  breadth  is  52920  rods  ;  what  is  the 
breadth  ?     How  many  acres  does  it  contain  ? 

3.  What  is  the  difference  between  five  hundred  sixty-nine 
thousandths,  and  five  hundred  sixty-nine  millionths  ? 

4.  A  man  having  nine-tenths  of  an  acre  of  land,  sold  nine- 
teen thousandths  of  an  acre;  how  much  did  he  have  left? 

5.  A  pile  of  wood  containing  150  cords  is  120  feet  long  and 
3|-  feet  wide ;  what  is  its  height  ? 

6.  Sold  96  yards  of  carpeting  at  $1.87|-  per  yard,  and  thereby 
gained  139 ;  how  much  did  it  cost  me  a  yard  ? 

7.  Change  22  years  122  days  to  days,  allowing  for  five  leap 
years. 

8.  What  is  the  cost  of  5  bu.  3  pk.  and  7  qt.  of  clover  seed,  at 
$4.85  per  bushel? 

9.  What  will  be  the  cost  of  fencing  a  lot  of  land,  120  rods  by 
260  rods,  at  37|-  cts.  per  foot  ? 

10.  The  product  of  four  numbers  is  6048,  and  three  of  the 
numbers  are  9,  12,  and  8  ;  what  is  the  other  factor? 

11.  A  man  sold  12  bu.  3  pk.  6  qt.  of  cranberries  at  $3^  a 
bushel,  and  took  his  pay  in  flour  at  4  cents  a  pound  ;  allowing 
196  lb.  to  a  bbl.,  how  many  barrels  should  he  receive  ? 

12.  How  many  steps  of  30  inches  must  a  j^erson  take  in  walk- 
ing 42  miles  ? 

13.  A  person  returning  from  the  mines  had  25  lb.  10  oz.  of 
pure  gold  ;  he  sold  it  at  ^1.044  per  pwt.     What  did  he  receive  ? 

14.  The  product  of  the  length,  breadth,  and  height  of  a  rect- 
angular hay -mow  is  3840  cubic  feet;  its  length  is  20  feet,  and 
its  breadth  is  16  feet.     What  is  its  height  ? 

15.  Bought  15  cwt.  22  lb.  of  rice  at  $4.25  a  cwt.,  and  6  cwt. 
36  lb.  of  pearl  barley  at  15.60  a  cwt.  ;  what  would  be  gained  by 
selling  the  whole  at  6 J  cts.  a  pound? 


340  Test  Questions, 

16.  From  a  piece  of  cloth  containing  seventy-five  and  seven- 
teen liundredths  yards,  thirty-six  and  seven  thousandths  yards 
were  used;  how  many  yards  were  left  ? 

17.  At  an  election  for  mayor,  8654  votes  were  cast  for  2  per- 
sons; the  successful  candidate  had  756  majority.  How  many 
votes  had  each  ? 

18.  A  lady  paid  11500  for  an  India  shawl  and  a  set  of  furs  ; 
the  difference  in  their  price  was  $575.  What  did  slie  pay  for 
each  .^ 

19.  The  sum  of  two  numbers  is  1876 ;  the  greater  is  3  times 
the  less.     What  are  the  numbers? 

20.  A  and  B  engaged  in  an  adventure  and  made  11575 ;  they 
divided  the  gain  in  such  a  manner  that  A  had  4  times  as  much 
as  B.     How  much  did  each  have  ? 

21.  The  sum  of  two  numbers  is  121^-;  their  difference  is  17^. 
What  are  the  numbers  ? 

22.  Find  the  greatest  common  divisor  and  least  common 
multiple  of  36,  79,  48,  and  69. 

23.  A  merchant  bought  360  yd.  of  silk  and  468  yd.  of  poplin, 
and  wished  them  cut  into  equal  dress  patterns  containing  the 
greatest  possible  number  of  yards;  how  many  patterns  could  he 
cut  from  both? 

24.  A  man  bought  3  tracts  of  land  containing  112,  144,  and 
176  acres,  respectively,  which  he  fenced  into  equal  fields  of  the 
greatest  possible  number  of  acres ;  how  many  acres  did  each 
contain  ? 

25.  The  Atlantic  Cable  cost  as  follows :  2500  miles  at  1485 
per  mile  ;  10  miles  deep  sea  cable,  at  $1450  per  mile ;  25  miles 
shore  ends,  at  $1250  per  mile.     What  was  the  total  cost? 

26.  AVhat  is  the  number  which  divided  by  453  gives  the  quo- 
tient 307  and  the  remainder  109? 

27.  \Yhat  is  a  prime  factor  ?     The  prime  factors  of  2366? 

28.  A  man  working  for  $2  a  day,  paying  $4  a  week  for  board, 
saved  $72  in  10  weeks  ;  how  many  week  days  was  he  idle? 

29.  Find  the  greatest  common  divisor  and  the  least  common 
multiple  of  195,  285,  and  315. 

30.  Divide  360  into  4  such  parts,  that  the  second  shall  be  2 
times  the  first,  third  3  times  the  firstj  fourth  4  times  the  first? 


Test  Questions.  341 

31.  Reduce  ^,  -^,  ||,  and  4|  to  the  least  common  deuomi- 
nator. 

32.  What  is  the  smallest  sum  with  which  I  cau  stock  a  farm 
with  sheep,  cows,  and  oxen,  investing  the  same  amount  in  each, 
and  paying  for  the  first  14,  for  the  second  130,  and  for  the 
third  $48  each  ? 

33.  From  sixteen  ten  thousandths  take  27  millionths,  and 
multiply  the  difierence  by  20.5. 

34.  Three  express  messengers  make  continuous  trips  between 
New  York  and  Washington,  one  of  whom  can  perform  the 
round  trip  in  16  hr.,  the  second  in  18  hr.,  the  third  in  20  hr. ; 
if  all  leave  New  York  at  the  same  time,  how  soon  will  they  all 
meet  there  again? 

35.  A  merchant  sold  3  pieces  of  broadcloth,  one  containing 
35|  yd.,  another  38|-  yd.,  another  42^^,  at  |5J  a  yard  ;  what  was 
the  amount  of  the  bill  ? 

36.  Wliat  is  the  total  of  the  following  bill :  3  dozen  eggs  at 
15  cents  a  dozen,  7  pounds  of  butter  at  23  cents  a  pound,  47 
yards  of  cotton  at  12  cents  a  yard,  and  8  pounds  of  coffee  at 
32  cents  a  pound  ? 

37.  A  farmer  sold  48J  bu.  potatoes,  at  62|  cts.  a  bushel,  and 
took  his  pay  in  coffee  at  33|-  cts.  a  pound ;  how  much  coffee 
did  it  require  to  pay  for  the  potatoes  ? 

38.  A  clerk  in  a  banking  house  spent  1475  for  house  rent, 
$350  for  clothing,  and  for,  family  expenses  $850,  the  sum  of 
which  was  -|  of  his  salary  ;  what  was  his  salary  ? 

39.  If  -f^  of  a  steamboat  cost  $9760,  what  is  the  whole  worth  ? 

40.  A  monument  standing  in  a  plane  cast  a  shadow  of  65 
feet,  which  was  -J  of  its  height;  what  was  the  height  of  the 
monument? 

41.  What  is  the  1.  c.  m.  of  f,  J,  f  ? 

42.  If  A  can  do  a  piece  of  work  in  7  days  which  A  and  B 
can  do  in  5  days,  in  how  many  days  will  B  do  the  same  work  ? 

43.  A  farmer  sells  7643  lb.  of  hay  at  $9.50  per  ton,  and  a  pile 
of  wood  6  feet  high,  11  feet  long,  and  4  feet  wide,  at  $3.50  per 
cord.     How  much  does  he  receive  for  both  ? 

44.  A  merchant  sells  cloth  at  $3.60  a  yard,  and  gains  20  per 
cent ;  for  what  price  must  he  sell  it  to  lose  15  per  cent  ? 


342  Test  Questions. 

45.  Divide  429  hundredths  by  5  millionths,  and  from  the 
quotient  subtract  425  thousandths;  express  the  result  ia  the 
lowest  terms  of  a  common  fraction. 

46.  What  will  be  the  cost  of  laying  a  pavement  30  feet  long 
and  8  feet  6  inches  wdde,  at  60  cents  per  square  yard  ? 

47.  What  sum  of  money  will  yield  as  much  interest  in  2  yr., 
at  10  per  cent,  as  $800  yields  in  5  yr.  3  mo.,  at  6  per  cent  ? 

48.  What  is  the  difference  between  the  true  and  the  bank 
discount  of  a  note  of  $600,  payable  in  40  days,  at  7  per  cent, 
without  grace  ? 

49.  A  sold  two  lots  for  $260  apiece,  gaining  20  per  cent  on 
one  and  losing  20  per  cent  on  the  other;  did  he  gain  or  lose, 
and  how  much  ? 

50.  A  commission  merchant  sold  500  pieces  of  cloth  for  $130 
a  piece,  and  paid  the  owner  $54,800;  what  was  the  rate  of  his 
commission  ? 

51.  How  much  will  it  cost  to  carpet  a  parlor  18  feet  square 
with  carpeting  -§  of  a  yard  wide,  at  $1.50  per  yard? 

52.  What  is  the  difference  between  the  market  value  and  the 
par  value  of  stock  ?     Between  a  dividend  and  an  assessment  ? 

53.  Two  men  start  from  the  same  point,  one  traveling  52 
miles  north,  the  other  39  west ;  how  far  apart  are  they  ? 

54.  If  eight  men  cut  84  cords  of  wood  in  12  days,  working 
7  hours  a  day,  how  many  men  will  it  take  to  cut  150  cords  in 
10  days,  working  5  hours  a  day  ? 

55.*^  Find  the  cube  root  of  42875000. 

56.  A  lady  bought  65  yards  of  dress  goods  at  25  cts..  If  yd. 
of  drilling  at  15  cts.,  13  yd.  of  cambric  muslin  at  12  cts.,  2  spools 
of  silk  at  20  cts.,  1  spool  of  cotton  at  5  cts.,  1  doz.  buttons  at 
25  cts. ;  write  the  bill  in  proper  form  to  show  that  it  is  paid, 
and  calculate  the  amount. 

57.  If  to  a  certain  number  you  add  its  half,  its  third,  and  28, 
the  sum  will  be  3  times  the  number  ;  what  is  the  number? 

58.  Wichita  is  40  miles  on  a  straight  line  directly  northwest 
of  Winfield ;  how  many  miles  will  a  person  travel  in  making 
the  jour^iey,  going  on  the  section  lines  ? 

59.  How  many  yards  of  carpeting  2 J  yards  wide  would  be  re- 
quired to  carpet  a  room  12^  yards  long  and  8f  yards  wide  ? 


Test  Questions.  343 

60.  Multiply  3  yr.  123  da.  5  lir.  17  min.  45  sec.  by  63. 

61.  A  grain  merchant  buys  at  different  times,  315  bu.  15  qt., 
843  bu.  19  qt.,  1243  bu.  'zi Q[t.,  and  734  bu.  7  qt.  of  oats,  at 
30  cents  per  bushel ;  how  much  money  did  he  pay  out  ? 

62.  What  will  9784  pounds  of  hay  cost,  at  110  per  ton  ? 

63.  In  making  j)urchases,  I  find  I  spend  -J  of  my  money  at 
the  first  store,  J  at  the  second,  and  \  at  the  third,  and  then 
have  $13  left;  how  many  dollars  had  I  at  first? 

64.  What  is  the  total  of  the  following  bill :  3^  yd.  at  $1.50, 

1  at  $1.62},  9f  at  $2.37|,  17}  at  85  cents  per  yard  ? 

65.  Demonstrate  that  when  we  multiply  the  numerator  of 
any  fraction  by  a  number,  we  increase  the  value  of  the  fraction 
as  many  times  as  there  are  units  in  the  multiplier. 

66.  Why  do  we  invert  the  divisor  in  the  division  of  one  frac- 
tion by  another  ? 

67.  Simplify  f  of  I  of  I  of  f  -=-  (5i  X  3}  X  4}  X  -|-)- 

68.  If  I  of  my  share  of  a  farm  is  worth  $510,  and  I  own  f  of 
the  farm,  what  is  the  value  of  the  farm  ? 

69.  What  number  must  be  divided  by  one-half  of  90  to  pro- 
duce three-fourths  of  228  ? 

70.  What  does  the  product  of  all  the  common  prime  factors 
of  two  or  more  numbers  produce  ? 

71.  If  two  men  are  50  miles  apart  and  travel  toward  each 
other,  one  going  3}  miles  per  hour,  and  the  other  3}  miles  per 
hour,  in  what  time  will  they  meet?  What  part  of  the  distance 
will  the  first  one  travel  ? 

72.  Multiply  seventy-eight  ten-thousandths  by  five  hun- 
dredths; divide  the  product  by  thirteen  thousandths,  and 
reduce  the  quotient  to  a  common  fraction  in  its  lowest  terms. 

73.  What  will  be  the  cost,  at  $0.50  per  cord,  of  a  load  of 
wood  consisting  of  two  lengths  of  four  feet  each,  the  load  being 

2  ft.  9  in.  wide  and  3  ft.  high  ? 

74.  If  a  man  uses  a  pound  of  fertilizers  on  a  piece  of  ground 
two  yards  square,  how  much  will  he  use  on  J  of  an  acre  ? 

75.  Five  cents  per  day  is  the  interest  on  what  sum  at  7  per 
cent  per  annum  ? 


344  Tent  Questions. 

76.  Write  a  negotiable  promissory  note,  observing  the  follow- 
ing directions:  date,  to-day;  face,  $150;  maker,  John  Jones; 
payee,  George  Green;  drawing  Q%  int.,  payable  in  4  montlis. 

77.  If  a  man's  property  is  assessed  at  $5125,  and  his  State 
tax  is  five  cents  on  a  thousand  dollars,  his  county  tax  one-half 
cent  on  a  dollar,  his  school  tax  three  mills  on  a  dollar,  and  his 
poll  tax  three  dollars,  Avhat  is  his  whole  tax  ? 

78.  A  pole  63  feet  long  was  broken  into  two  pieces,  the 
shorter  being  -|  of  the  longer.     Required  the  length  of  each. 

79.  A  can  hoe  a  row  of  corn  in  a  certain  field  in  30  minutes, 
B  can  hoe  a  row  in  20  minutes,  and  0  can  hoe  a  row  in  35 
minutes.  What  is  the  least  number  of  rows  that  each  can  hoe 
in  order  that  all  may  finish  together  ? 

80.  What  number  is  that  from  which  if  you  take  ^  of  itself, 
3f  times  the  remainder  minus  1  will  be  50  ? 

81.  What  is  the  amount  due  for  the  following: 

700^^  feet  of  boards  @  122.50  per  M.  ; 
912  pounds  of  hay  @  $14.50  per  ton? 

82.  Philip  Davis  is  debtor  to  William  Richmond,  Albion,  as 
follows:  For  16J  yards  sheeting  at  22  cents  per  yard,  7|-  yards 
flannel  at  G2-J-  cents  per  yard,  ^  dozen  handkerchiefs  at  37-|- 
cents  each,  and  2|-  yards  drilling  at  15|-  cents  per  yard.  The 
above  bill  was  paid  November  23,  1881.  Make  out  a  receipted 
bill  in  proper  form. 

83.  A  buys  a  horse  for  $60,  and  sells  it  to  B  for  $120,  who 
sells  it  for  $200 ;  what  was  the  difference  in  their  per  cent  of 
gain  ? 

84.  Had  the  cost  price  of  an  article  been  twenty  per  cent 
less,  the  rate  of  loss  had  been  fifteen  per  cent  less ;  what  wa3 
the  rate  of  loss  ? 

85.  A  hired  of  B  $1000  for  one  3'ear,  at  12  per  cent  in  ad- 
vance, and  gave  his  note  for  the  $1000,  paying  $120  down.  At 
the  end  of  the  year  A  said  to  B,  "  I  want  the  $1000  another 
year  on  the  same  terms."  "  Well,"  says  B,  "  give  me  the  $120." 
A  gives  him  a  check  for  $300,  saying,  "Take  out  the  interest, 
and  indorse  the  rest  on  the  new  note."  How  much  should  B 
indorse  on  the  note  ? 


Test  Questions,  345 

86.  A  Syracuse  coal  dealer  bought  (at  wholesale)  at  a  mine 
iu  Penn.  1540  tous  anthracite  coal  at  13.50  per  ton.  The 
freight  to  Syracuse  was  $2040,  and  the  loss  in  transportaliou 
was  estimated  at  15510.  The  coal  was  retailed  at  85.50  per  ton. 
What  was  the  gain  ? 

87.  Sold  5000  pounds  of  sugar  at  9  cents  per  pound,  and  lost 
10  per  cent ;  what  per  cent  should  1  gain  by  selling  it  at  12 
cents  per  pound  ? 

88.  Three  persons  purchased  a  horse  together.  A  gave  S20, 
B  gave  40  per  cent  more  than  A,  and  C  gave  15J  per  cent  less 
than  both  the  others.  What  fractional  part  of  the  horse  does 
each  own  ? 

89.  A  man  bought  1000  bushels  of  wheat  for  11250.  He 
finds  lb%  of  it  worthless.  For  how  much  must  he  sell  the  re- 
mainder per  bushel  to  gain  20^^'  on  the  cost? 

90.  At  what  rate  per  cent  must  I  invest  $600,  that  in  2  yr. 
6  mo.  it  may  amount  to  $?05  ? 

91.  For  what  sum  mnst  a  note  be  written  in  order  to  receive 
from  a  bank  $540  at  Q%  for  60  days  ? 

92.  If  6  men  dig  a  trench  15  yards  long,  4  yards  broad,  and 
5  feet  deep,  in  3  days  of  12  hours,  in  hoAv  many  days  of  8  hours 
will  8  men  dig  a  trench  20  yd.  long,  8  yd.  broad,  and  8  ft.  deep? 

93.  What  sum,  put  at  simple  interest  at  1^%  per  annum, 
will  amount  in  3  yr.  4  mo.  to  82500  ? 

94.  The  interest  on  a  certain  sum,  for  2J  years  at  7;^, 
is  85.87-|.  What  is  the  true  discount  on  the  same  sum  for  the 
same  time,  at  the  same  rate  ? 

95.  A  merchant  bought  a  certain  number  of  yards  of  cloth  at 
S2.50  ]3er  yard.  He  sold  two-fifths  of  the  cloth  at  a  profit  of 
25;^,  and  on  the  sale  of  the  remainder  he  lost  115.  If  his  loss 
on  the  whole  transaction  amounted  to  bfc,  how  many  yards  of 
cloth  did  he  buy  ? 

96.  If  it  cost  $95.60  to  caq^et  a  room  24  x  18  ft,,  how  much 
will  the  same  kind  of  carpet  cost  for  a  room  38  x  22  ft.  ? 

97.  What  sum  of  money  is  that  of  which,  if  80^'  be  deposited 
in  bank,  and  20;;^'  of  this  deposit  be  withdrawn,  there  will  re- 
main 15760  in  bank? 


346  T€8t  Questions, 

98.  A  lawyer  collecting  a  note  at  a  commission  of  %%,  re- 
ceived 16.80  ;  what  was  the  face  of  the  note? 

99.  Bought  stock  at  par,  and  sold  it  at  3%  premium,  thereby 
gaining  1750  ;  how  many  shares  of  1100  each  did  I  buy  ? 

100.  What  is  the  amount  of  $16941.20  for  1  yr.  7  mo.  28  da., 
at  4:l^/c,  simple  interest. 

101.  An  investment  of  17266.28  yields  1744.7937  annually ; 
what  is  the  rate  of  interest  ? 

102.  In  what  time  will  1273.51  amount  to  $312864,  at  7^, 
simple  interest  ? 

103.  What  is  the  difference  between  the  interest  and  the  true 
discount  of  1576,  due  1  yr.  4  mo.  hence,  at  Q)%  ? 

104.  Three  men  gain  $2640,  of  which  B  is  to  have  16  as  often 
as  C  14  and  A  12 ;  what  is  each  one's  share  ? 

105.  Find  the  square  root  of  10795.21  to  three  decimals. 

106.  What  is  the  length  of  one  side  of  a  square  piece  of  land 
containing  40  acres  ? 

107.  A  room,  the  height  of  which  is  11  feet  and  the  length 
twice  the  breadth,  takes  143  yards  of  paper  2  feet  wide  to 
cover  its  walls,  door  and  windows  included ;  how  many  yards 
of  carpet  27  inches  wide  will  be  required  for  the  floor  ? 

108.  In  a  rectangular  cistern  the  length  is  12  feet,  the  width 
is  5  ft.,  and  depth  3  ft. ;  find  the  diagonal  through  the  centre  of 
the  rectangular  space.  Find  the  weight  of  water  in  pounds  it 
will  contain,  if  a  cubic  foot  of  water  weighs  1000  ounces. 

109.  A  man  sawed  a  pile  of  wood  40  ft.  long,  4  ft.  wide, 
5J  ft.  high,  for  11.50  per  cord.     How  much  did  he  earn  ? 

110.  What  will  be  the  cost  of  35  three-inch  planks  22  ft.  long, 
16  inches  wide,  at  $17.50  per  M.  ? 

111.  How  many  bushels  will  a  bin  hold  that  is  9  ft.  long, 
6  ft.  wide,  6  ft.  high  ? 

112.  A  note  was  given  Jan.  1,  1880,  for  $700.  The  follow- 
ing payments  were  indorsed  upon  it :  May  6, 1880,  $85 ;  July  1, 

1881,  $40  ;  Aug.  20,  1881,  $100.     How  much  was  due  Jan.  10, 

1882,  interest  at  6  per  cent? 

113.  At  what  price  must  5  per  cent  bonds  be  bought  so  as  to 
realize  7  per  cent  on  the  investment  ? 


Test  Questions.  347 

114.  Wliat  will  be  the  cost  in  Buffalo,  N.  Y.,  of  a  draft  for 
$1500  on  Cleveland,  0.,  payable  90  days  after  date,  exchange 
■J  per  cent  discount  ? 

115.  If  I  place  $1500  at  interest  for  18  months,  and  receive 
1135  interest,  what  sum  must  I  place  at  interest  at  the  same 
rate,  that  I  may  receive  $275  interest  in  8  months? 

116.  The  length  of  a  rectangular  field  containing  20  acres  is 
twice  its  width;  what  is  the  distance  around  it  ? 

117.  Find  the  amount  of  $387.20,  from  Jan.  1  to  Oct.  20, 
1881,  at  7  per  cent. 

118.  A  man  was  offered  $3675  in  cash  for  his  house,  or  $4235 
in  three  years  without  interest.  He  accepted  the  latter  offer; 
did  he  gain  or  lose,  and  how  much,  money  being  worth  7  per 
(Cei3 1  ? 

119.  What  are  the  proceeds  of  a  note  for  $368,  at  90  days, 
discounted  at  bank  at  6  j^er  cent  ? 

120.  The  height  of  the  Obelisk,  known  as  Cleopatra's  needle, 
at  N.  Y.  Central  Park,  is  70  feet,  nearly ;  the  diameter  of  the 
base  is  about  8  feet.  What  is  the  length  cf  a  line  drawn  from 
the  apex  to  a  point  36  ft.  from  the  middle  of  one  side  of  the 
base  ? 

121.  A  ship  sails  east  from  Boston,  long.  71°  10',  at  the  rate 
of  2°  30'  20"  in  a  day ;  what  is  her  long,  at  the  end  of  6  days  ? 

122.  If  a  man  wastes  5  minutes  a  day,  how  much  time  will 
he  waste  in  a  common  year? 

123.  How  many  cu.  ft.  of  water  must  be  drawn  from  a  reser- 
voir 30  ft.  6  in.  long  and  20  ft.  6  in.  wide,  to  lower  the  surface 
8  inches  ? 

124.  The  Signal  Service  reports  that  3|  in.  of  rain  fell  in 
24  hours ;  how  many  cu.  yd.  fell  on  an  acre  of  ground  ? 

125.  What  is  the  difference  between  35  square  rods  and  35 
rods  square  ? 

126.  If  a  bird  can  fly  12J  miles  in  \  hour,  how  far  can  it  fly 
in  b^  hours  ? 

127.  If  3  cheeses  wxigh  35|,  44y\,  and  27i  lb.,  what  is  their 
entire  w^eight  ?     W^hat  is  their  average  weight  ? 

128.  If  4  is  added  to  both  terms  of  the  fraction  |,  will  the 
yalue  be  increased  or  diminished,  and  how  much  ? 


348  Test  Questions, 

129.  A  lad  having  3  quarts  of  berries,  ate  J  of  them,  sold  \ 
of  the  remainder,  and  divided  the  rest  between  his  two  com- 
panions ;  how  many  did  each  receive  ? 

130.  My  gas  bill  was  $12  when  I  burned  4800  feet  of  gas; 
what  will  it  be  when  gas  costs  ^  more,  and  I  burn  1600  ft.  less? 

131.  The  difference  in.  time  between  Greenwich  and  St.  Louis 
is  5  hr.  and  55  min. ;  what  is  the  difference  in  longitude  ? 

132.  How  many  cubic  feet  in  10  boxes,  each  7J  ft.  long.  If 
ft.  wide,  and  IJ  ft.  high  ? 

133.  If  y9^  of  a  saw-mill  are  worth  1631.89,  what  are  -{^  of  it 
worth  ? 

134.  Find  the  difference  in  time  between  two  places  whose 
difference  of  longitude  is  5°  40'. 

135.  The  Hoosac  Tunnel  is  25000  feet  long ;  Mount  Cenis 
Tunnel,  which  connects  France  and  Italy,  is  12  kilometers. 
What  fraction  of  the  latter  is  the  former  ? 

136.  A  broker  received  125250.50  to  invest  in  stocks,  after 
deducting  his  commission  of  2|-^ ;  what  was  his  commission, 
and  how  much  did  he  invest  ? 

137.  A  grain  dealer  sent  a  boat-load  of  corn  to  market,  valued 
at  $2000,  and  insured  62-|-;^  of  its  value  at  1^%',  what  premium 
did  he  pay  ? 

138.  If  I  sell  wood  at  17.20  per  cord,  and  gain  20  per  cent, 
what  did  it  cost  me  per  cord  ? 

139.  If  5  men  can  harvest  a  field  in  12  hours,  how  many 
hours  would  it  require  if  4  more  men  were  employed? 

140.  If  15  oxen  or  20  horses  eat  6  tons  of  hay  in  8  weeks, 
how  much  will  12  oxen  and  28  horses  require  in  21  Aveeks? 

141.  What  must  be  the  depth  of  a  cubical  cistern  that  will 
hold  3048.625  cubic  feet  of  water? 

142.  How  many  tiles  8  in.  square  will  cover  a  floor  18  ft.  long 
and  12  ft.  wide  ? 

143.  What  per  cent  of  a  mile  is  150  rd.  2  yd.  2J  ft.  ? 

144.  The  assets  of  a  bankrupt  are  165000,  and  his  liabilities 
$85500  ;  what  per  cent  can  he  pay  ? 

145.  A  merchant  reduced  tlie  price  of  a  piece  of  cloth  15  cts. 
per  yd.;  and  thereby  reduced  his  profit  on  the  cloth  from  12  to 
%% ;  what  was  the  cost  of  the  cloth  ? 


Te8t  Questions.  '  349 

146.  How  many  feet  are  there  iu  Qb%  of  a  mile  ? 

147.  A  man  whose  income  was  1^700  spent  11500 ;  what  per 
cent  of  his  income  remained  ? 

148.  Paid  ^8000  for  stock  1%%  below  j^ar,  and  sold  it  at  115  ; 
what  per  cent  did  I  gain  ? 

149.  If  A  loans  B  $500  for  6  mo.,  how  long  ought  B  to  lend 
A  $800  to  requite  the  favor? 

150.  The  ratio  is  3|,  the  antecedent  J  of  |;  what  is  the  con- 
sequent ? 

151.  How  long  will  it  take  12  men  to  do  a  job  which  7  men 
can  do  in  15|-  days? 

152.  What  is  the  difference  between  the  cube  and  ihe  cube 
root  of  .027  ? 

153.  What  are  the  dimensions  of  a  cube  equal  to  a  bin  12  ft. 
6  in.  long,  10  ft.  wide,  and  5  ft.  high  ? 

154.  The  longitude  of  Boston  is  71°  10'  W.,  and  that  of  New 
Orleans  is  90°  2'  W. ;  what  is  the  time  at  New  Orleans  when  it 
is  7  o'clock  12  min.  a.  ai.  at  Boston  ? 

155.  AVhat  will  be  the  wages  of  9  men  for  11  days,  if  the 
wages  of  6  men  for  14  days  be  $84  ? 

156.  For  how  much  must  I  make  my  note  at  bank  for  3  mo. 
at  Q%,  in  order  to  get  from  the  bank  just  $300  ? 

157.  Bought  a  horse  for  $125,  and  sold  it  for  20^  advance; 
sold  a  carriage  for  $125,  gaining  25^^' ;  sold  a  yoke  of  oxen  for 
$125,  losing  20^/  ;  bought  ten  sheep  for  $125,  and  sold  them  at 
a  loss  of  25C/.     What  did  I  gain  or  lose  on  the  whole? 

158.  Of  two  pieces  of  land,  the  one  a  circle  18  rods  in  diam- 
eter, the  other  a  triangle  whose  hypothenuse  is  30  rods,  and 
whose  base  is  24  rods,  which  is  the  larger,  and  how  much? 

159.  The  length  of  a  block  of  marble  containing  105  cu.  in. 
is  7  inches;  find  the  length  of  a  similar  block  containing  22680 
cubic  inches. 

160.  The  sum  of  two  fractions  is  -fj-l,  and  their  difference  -^', 
wiiat  are  the  fractions? 

161.  A  person  expended  1Q%  of  all  he  was  worth  in  buying 
20^  of  the  stock  of  a  mining  company.  If  the  entire  stock  of 
the  company  sold  for  $100000,  w4iat  was  the  person  worth  ? 


350  Test  Questions. 

162.  A  trader  sold  75  cords  of  wood  for  $487.50,  thereby  los- 
ing 10;^  of  the  cost;  what  did  the  wood  cost  per  cord? 

163.  Simplify  —  -f  —  +  — . 

«  ^  «» 

164.  Gunpowder  being  f  nitre  and  equal  parts  of  sulphur 
and  charcoal,  how  many  pounds  of  each  of  the  three  are  there 
in  a  ton  ? 

165.  Extract  the  square  root  of  1.225784  to  four  decimals. 

166.  Divide  $4600  into  parts  which  are  to  each  other  as  ^y  f, 
and  |. 

167.  What  capital  must  be  invested  in  5  per  cents  at  95,  to 
secure  an  income  of  $10000  ? 

168.  Find  the  present  worth  of  a  note  for  $175,  payable  in 
8  mo.,  interest  being  computed  at  7^  ? 

169.  }  is  what  per  cent  of  |  ? 

170.  If  a  merchant  sells  goods  which  cost  him  $1620  for 
$1800  on  a  9  mo.  credit  without  interest,  money  being  worth  6 
per  cent,  how  much  does  he  gain  ? 

171.  A  furrier  asked  40^  more  for  a  set  of  mink  furs  than 
they  cost  him ;  but  he  afterwards  sold  them  at  a  reduction  of 
10^^  from  the  price  asked,  thus  realizing  from  the  sale  $12.22 
profit.     What  did  the  furs  cost  him  ? 

172.  In  what  time  will  $560,  at  8^  per  annum,  produce 
$106.40  interest  ? 

173.  I  owe  a  debt  of  $325.50,  due  in  1  yr.  5  mo.,  without  in- 
terest ;  what  will  pay  the  debt  now,  money  being  worth  Q%  per 
annum  ? 

174.  If  15  men,  working  6  hours  a  day,  can  dig  a  cellar  80  ft. 
long,  60  ft.  wide,  and  10  ft.  deep,  in  25  da.,  how  many  days  will 
it  take  25  men,  working  8  hr.  a  day,  to  dig  a  cellar  120  ft.  long, 
70  ft.  wide,  and  8  ft.  deep  ? 

175.  If  by  selling  a  liouse  for  $12500  a  builder  gains  25^, 
what  per  cent  would  he  gain  or  lose  by  selling  it  for  $9000  ? 

176.  What  is  the  area  of  a  right-angled  triangle,  whose  per- 
pendicular is  32  ft.,  and  its  hypothenuse  40  ft.  ? 

177.  Find  the  value  of  ^r0()0238328. 

178.  How  much  must  be  paid  for  making  52  rd.  14  ft.  8  in. 
of  fence,  at  $.  75  per  foot  ? 


Test  Questions.  351 

179.  A  traveler,  on  reaching  a  certain  place,  found  that  his 
watch,  which  gave  the  correct  time  for  the  place  he  left,  was 
2  hr.  22  min.  slower  than  the  local  time.  Had  he  traveled  east- 
ward or  westward,  and  how  many  degrees  ? 

T.-    1  n  ^  -06  +  .03-1        ,  4  of  2i 

180.  Find  the  sum  of  -— — -—^  and  ^^— — -• 

"^5  —  ^2  '^   +  "8 

181.  What  number  diminished  by  36^  of  itself  =  336? 

182.  What  is  the  value  of  a  meadow  70  rd.  long  and  20  rd. 
wide,  at  147.25  per  acre? 

183.  What  is  the  area  of  a  triangle  whose  sides  are  respectively 
10  ft.,  12  ft.,  and  16  ft.? 

184.  What  is  the  area  of  a  triangle  whose  sides  are  each 
24  yards? 

185.  A  man  bought  a  garden  3  rods  wide  and  4  rods  long, 
and  agreed  to  pay  1  cent  for  the  first  square  rod,  4  cents  for  the 
second,  16  cents  for  the  third,  and  so  on,  quadrupling  each  sq. 
rod ;  how  much  did  his  garden  cost  him  ? 

186.  Fiud  the  balance  due  March  4,  1882,  on  a  note  dated 
Jan.  1,  1879,  for  |>580  at  b%,  on  w^hich  a  payment  of  185  had 
been  made  every  6  months ;  using  the  U.  S.  rule. 

187.  A  and  B  enter  into  partnership  ;  A  furnished  $240  for 
8  mo.,  and  B  1559  for  5  mo.  They  lost  $118  ;  how  much  did 
each  man  lose  ? 

188.  In  25  kilogrammes  how  many  pounds,  Troy  weight  ? 

18  -^  1 

189.  Reduce  — — '—^  to  its  simplest  form. 

190.  What  is  the  area  in  acres  of  a  triangle  whose  base  is 
156  rods  and  its  altitude  63  rods  ? 

191.  Suppose  a  certain  township  is  6  miles  long  and  4|-  miles 
wide,  how  many  lots  of  land  of  90  acres  each  does  it  contain  ? 

192.  How^  many  strokes  would  a  clock  wiiich  goes  to  24 
o'clock,  strike  in  a  day  ? 

193.  The  extremes  are  3  and  19,  the  number  of  terms  9; 
what  is  the  com.  dif.^  and  the  sum  of  the  series  ? 

194.  A  man  spent  13  the  first  holiday,  l?45  the  last,  and  each 
day  $3  more  than  on  the  preceding  ;  how  many  holidays  did  he 
have,  and  how  much  did  he  spend  ? 

195.  What  is  the  area  of  a  circle  whose  diameter  is  120  rd.  ? 


352  Test  Questiom. 

196.  How  much  should  be  discounted  on  a  bill  of  $3725.87, 
due  in  8  mo.  10  da.,  if  paid  immediately,  money  being  worth 
5  per  cent  ? 

197.  Bouglit  bonds  at  115  aiid  sold  at  110,  losing  $300.  How 
many  bonds  of  $1000  each  did  I  buy? 

198.  What  is  the  amount  of  1225,  at  6  per  cent  compound 
interest  for  4  years  ? 

199.  A  steamer  goes  due  north  at  the  rate  of  15  miles  an 
hour,  and  another  due  west  18  miles  an  hour ;  how  far  apart 
will  they  be  in  24  hours  ? 

200.  Find  the  cost,  at  30  cts.  per  sq.  yd.,  of  plastering  the 
bottom  and  sides  of  a  cubical  cistern  that  will  hold  300  bbls. 

201.  Find  the  surface  and  the  diagonal  of  a  cube  of  granite 
containing  162144  cu.  inches. 

202.  What  is  the  area  of  a  circle  whose  circumference  is  160 
yards  ? 

203.  What  is  the  solidity  of  a  prism  whose  height  is  25  ft., 
and  its  base  an  equilateral  triangle  whose  side  is  12  feet? 

204.  What  is  the  solidity  of  a  prism  whose  base  is  6  ft.  square 
and  its  height  15  feet? 

205.  What  is  the  solidity  of  a  triangular  prism  whose  height 
is  20  feet,  and  the  area  of  whose  base  is  460  square  feet? 

206.  Required  the  solidity  of  a  square  pyramid,  the  side  of 
whose  base  is  25  feet,  and  whose  height  is  ijQ  feet. 

207.  Required  the  solidity  of  a  cone,  the  diameter  of  whose 
base  is  30  feet,  and  whose  height  is  96  feet. 

208.  Required  the  solidity  of  a  cylinder  20  feet  in  diameter 
and  65  feet  long. 

209.  How  many  acres  in  a  triangular  field  whose  base  is 
325  yd.  and  its  altitude  160  yd.  ? 

210.  If  a  scholar  receive  1  credit  mark  for  the  first  example 
he  solves,  2  for  the  second,  4  for  the  third,  and  so  on,  the  num- 
ber being  doubled  for  each  example,  how  many  marks  will  he 
receive  for  the  twelfth  ? 

211.  What  rate  of  income  will  U.  S.  34^  bonds  yield,  if 
bought  at  102,  and  payable  at  par  in  25  years? 

212.  What  per  cent  income  will  Alabama  9's  yield,  bought 
at  85  and  paid  at  par  in  15  years  ? 


Test  Questions,  353 

III.    Problems 
FROM  Entrance  Examination  Papers  of  Various  Colleges. 

Harvard  University,  1880,  '81. 

859.    1.  Find  the  greatest  common  divisor  of  315,  504,  441. 

2.  Find  the  square  root  of  2  to  the  nearest  ten-thousandth. 

3.  A  wall  which  was  to  be  36  feet  high  was  raised  9  feet  in 
6  days  by  16  men  ;  how  many  men  will  be  needed  to  finish  the 
w^ork  in  4  days  ? 

4.  A  tradesman  marks  his  goods  at  25  per  cent  above  cost, 
and  deducts  12  per  cent  of  the  amount  of  any  customer's  bill, 
for  cash.     What  per  cent  does  he  make  ? 

5.  A  tunnel  is  2  miles  21  chains  13.2  yards  long.     Find  its 

length  in  meters.     [1  mile  =  1.61  kilometres.] 
1  7_5 q  3.  _|_  45. 

6.  Simplify  lliV-^^^- 

7.  Find  the  value  in  cubic  decimeters  of  {-|-  of  87  cu.  meters 
62  cu.  decimeters  300  cu.  centimeters. 

8.  If  27  men,  working  10  hours  a  day,  do  a  piece  of  work  in 
14  days,  how  many  hours  a  day  must  12  men  work,  to  do  the 
same  amount  in  45  days  ? 

9.  What  sum  of  money,  at  6  per  cent  annually  compounded 
interest,  will  amount  to  $2703  in  1  yr.  4  months? 

10.  Arrange  in  order  of  magnitude,  fl,  ^,  0.89. 

Yale  College,  1880. 

11.  Add  (fxixl),  tV  f,  and  tV 

12.  Divide  (p  -  ^)  by  A. 

13.  Find  the  fourth  term  of  a  proportion  of  which  the  first, 
second,  and  third  terms  are,  respectively,  3.81,  0.056,  1.67. 

14.  Reduce  133  sq.  rd.  8  sq.  ft.  to  a  decimal  of  an  acre. 

15.  In  a  board  4  meters  long  and  0.4  meters  wide,  how  many 

square  decimeters  ? 

3A 

16.  Divide  H  oi -{-^  oi  ^)  bv  ^-,  and  add  the  quotient  to 

4         iF' 


354  Test  Questions, 

17.  Find  V-Ti:,  to  three  decimal  places. 

18.  Find,  to  three  decimal  places,  the  number  which  has  to 
0.649  the  same  ratio  which  58  has  to  634. 

19.  A  man  bought  a  piece  of  ground  containing  0.316  A.  at 
53  cents  a  square  foot ;  what  did  he  pay  for  the  piece  ? 

20.  A  grocer  buys  sugar  at  18  cents  a  kilo,  and  sells  it  at  1 
cent  per  50  grams  ;  how  much  per  cent  does  he  gain  ? 

Columbia  College,  1881. 

21.  Define  a  fraction.  Give  the  rule  for  the  addition  of  frac- 
tions, and  the  reason  for  each  step  of  the  operation. 

22.  Eeduce  126  grams  to  ounces.     63f  yd.  to  meters. 

23.  From  f  of  a  gallon  take  If  of  a  pint.  What  difference, 
if  any,  between  the  subtraction  of  compound  numbers  and  that 
of  simple  ones  ?  Between  the  subtraction  of  fractions  and  that 
of  integers  ? 

24.  If  30  lb.  of  cotton  will  make  3  pieces  of  muslin  42  yd. 
long  and  f  yd.  wide,  how  many  pounds  will  it  take  to  make  50 
pieces,  each  containing  35  yd.,  1\  yd.  wide? 

25.  A,  B,  and  0  formed  a  partnership,  and  cleared  112000. 
A  put  in  $8000  for  4  mo.,  and  then  added  12000  for  6  mo.;  B 
put  in  $16000  for  3  mo.,  and  then  withdrawing  half  his  capital, 
continued  the  remainder  5  mo.  longer ;  0  put  in  $13500  for 
7  mo.     How  divide  the  profit  ? 

26.  Find  the  sum  of  3^,  6f,  8j^,  65f,  reduce  the  fractional 
part  to  a  decimal,  and  extract  the  cube  root  of  the  result. 

Dartmouth  College,  1879,  '80. 

27.  Find  the  I,  C  7n.  and  the  g,  c,  d,  of  6,  8,  20,  and  36. 

28.  How  many  metres  in  25  feet  ? 

29.  Find  the  square  root  of  3530641. 

30.  Gold  was  quoted  at  |1.12|-;  what  was  a  %1  greenback 
worth  ? 

31.  11200  includes  a  sum  to  be  invested  and  a  commission  of 
h%  of  the  sum  invested;  what  is  the  sum  invested? 

32.  Find  the  sum  and  product  of  -|,  J,  f . 

33.  Find  the  cube  root  of  3845672000. 


Test  Questions,  355 

34.  Find  the  square  root  of  3534400.5. 

35.  A  platl'orni  bears  a  weight  of  100  lb.  per  square  foot ; 
what  is  the  weight  in  kilograms  per  square  meter? 

36.  A  horse  that  cost  6^  per  cent  of  $25000,  was  sold  for 
$1000  ;  what  was  the  loss  per  cent  ? 

College  of  City  of  New  York,  1880,  '81. 

37.  Eeduce  f  of  ^  of  -y-  to  a  decimal,  carrying  out  the  opera- 
tion to  four  places. 

38.  If  two  men,  working  8  hours,  can  carry  12000  bricks  to 
the  height  of  50  fe^t,  how  many  bricks  can  one  man,  working 
10  hours,  carry  to  the  height  of  30  feet  ? 

39.  I  buy  goods  to  the  amount  of  $4,978.70,  payable  in  4  mo., 
with  interest  at  b^^,  and  give  my  note  without  interest.  What 
must  be  the  face  of  the  note  ? 

40.  A  man  lost  ^,  \,  and  f  of  his  money,  and  then  had  $2600 
left ;  what  sum  had  he  originally,  and  how  much  per  cent  had 
he  lost? 

41.  Sold  a  fire  engine  for  $7050,  and  lost  %%  on  its  cost ;  for 
how  much  ought  I  to  have  sold  it  to  gain  12;;^^'  ? 

42.  What  sum  of  money  put  at  interest  6  yr.  5  mo.  11  da., 
at  7^,  will  gain  $3159.14? 

43.  For  what  sum  must  a  note  be  drawn  at  60  days  to  net 
$1200  when  discounted  at  b%  ? 

44.  Extract  the  square  root  of  3286.9835  to  the  fourth  deci- 
mal place. 

45.  Extract  the  cube  root  of  30.625. 

Amherst  College,  1881. 

46.  Find  the  greatest  common  divisor  of  1263  and  1623. 

47.  Find  the  least  common  multii)le  of  18,  24,  36,  and  126. 

48.  A  cable  that  weighs  one  ton  per  mile  weighs  how  much 
per  foot  ? 

49.  When  it  is  10  o'clock  in  Boston  what  time  is  it  in  Am- 
herst, the  longitude  of  Boston  being  71°  7'  45"  W.  from  Green- 
wich, that  of  Amherst  being  72^  31'  50"  ? 

50.  Reduce  1  hr.  25  min.  30  sec.  to  the  decimal  of  a  day. 

51.  Of  what  number  is  f  the  J  part? 


356  Test  Questions. 

52.  What  must  be  the  face  of  a  note  which  discounted  at  a 
bank  at  Q%  for  30  days  and  grace,  would  yield  $200  ? 

53.  Sold  a  house  for  $5000  and  thereby  gained  20^.  Should 
I  have  gained  or  lost,  and  how  much  per  cent,  if  I  had  sold  it 
for  $4000  ? 

54.  Find  the  square  root  of  5.6169. 

55.  The  meter  is  39.37  inches.  Find  how  many  kilometers 
there  are  in  a  mile. 

Vassar  College,  1880. 

5e.  Add  ^^1  to  4^f . 

•5  +  1     ^  —  ^ 

57.  Multiply  48  ten  thousandths  by  two  and  one  thousandth, 
and  divide  the  result  by  one  million. 

58.  Express  462  mm.  in  higher  denominations. 

59.  What  is  1%  of  140  books?  What  per  cent  of  30  ft.  is 
25  inches  ? 

60.  If  I  lose  10;^  by  selling  goods  at  28  cents  per  yard,  for 
what  should  they  be  sold  to  gain  20%'? 

61.  What  principal  will  yield  an  interest  of  $339.20  in  5  yr. 
4  mo.  at  Q%  ? 

62.  What  must  be  the  length  of  a  box,  1  meter  wide  and 
1  meter  deep,  to  contain  4500  liters  ? 

63.  Cube  .01.     Square  1.001. 

64.  Extract  the  square  root  of  4.932841. 

65.  A  can  do  a  piece  of  work  in  10  days;  A  and  B  can  do 
the  same  work  together  in  7  days;  in  how  many  days  can  B 
working  alone  do  the  Avork  ? 

New  York  Normal  College,  1880. 

66.  What  will  it  cost  to  floor  a  room  17^  ft.  long  and  16  ft. 
wide,  at  the  rate  of  $1.10  per  sq.  yd.  ? 

67.  A  man  has  a  capital  of  $12500  ;  he  puts  15;^  of  it  in 
stocks,  33|C^  in  land,  and  25^^  ni  mortgages ;  how  many  dollars 
has  he  left  ? 

68.  A  grocer  bought  500  bags  of  coffee,  each  bag  containing 
49^  pounds,  at  12  cents  a  pound,  and  sold  at  a  profit  of  16-|%' ; 
for  what  did  he  sell  it  ? 


Test  Questions,  357 

69.  If  I  buy  a  house  for  $5620  and  receive  $1803  for  rent  in 
2  yr.  3  mo.  15  da.,  what  rate  of  int.  do  I  get  for  my  money? 

70.  Find  the  face  of  a  note  payable  in  90  da.  at  1i%,  so  that 
the  proceeds  shall  be  12050  ? 

71.  A  merchant  owes  12400,  of  which  $400  is  payable  in 
6  mo.,  $800  in  10  mo.,  and  $1200  in  16  mo. ;  what  is  the  equa- 
ted time  ? 

72.  If  it  costs  17.20  to  transport  18 J  cwt.  5 J  miles,  what  will 
it  cost  to  transport  112  J  tons  62|-  miles  ? 

73.  Extract  the  square  root  of  1051  to  three  places  of  deci- 
mals. 

74.  What  is  the  cube  root  of  403583419  ? 

Cornell  University,  1880, 

75.  What  is  the  value  of  50  lb.  8  oz.  of  gold,  at  $20.59;|-  per 
ounce  ? 

76.  Given  the  metre  equal  to  39.37  inches,  reduce  one  mile 
to  kilometers.     Give  the  metric  table  of  weights. 

77.  Divide  f  of  7|  by  f  of  12^.  Prove  the  result  by  reduc- 
ing the  fractions  to  decimals  and  working  the  example  anew. 

78.  How  long  must  $125  be  on  interest  at  7^  per  cent  to 
gain  $15  ? 

79.  Eeceived  6  per  cent  dividend  on  stock  bought  at  25  j^er 
cent  below  par ;  what  rate  of  interest  did  the  investment  pay  ? 

80.  How  many  liters  in  20  bu.  3  pk.  4  qt.,  the  bushel  being 
2150.42  cubic  inches,  and  the  metre  39.37  inches? 

81.  Simplify  (l  +  ^^)  -  (l  +  ^). 

82.  If  one  kilometer  equals  five-eighths  of  a  mile,  how  many 
turns  will  a  wheel  make  in  20  miles,  the  circumference  of  the 
wheel  being  4  meters  5  millimeters  ? 

83.  What  is  the  difference  between  the  true  and  bank  dis- 
count of  $250,  due  10  mo.  hence,  at  7f/  ? 

84.  If  8  men  spend  $32  in  13  weeks,  what  will  24  men  spend 
in  52  weeks  ? 


358  Te8t  Questions, 

Trinity  College,  1880. 

85.  Subtract  thirty  million  twenty-six  thousand  three  from 
45007021.  Find  what  number  must  be  added  to  the  difference 
to  make  one  hundred  million,  and  write  the  answer  in  words. 

86.  The  sum  of  |  and  -^j  is  diminished  by  ^.  How  many 
times  does  the  difference  contain  y\  of  the  sum  of  \,  ^,  and  yV-^ 

87.  Divide  375  by  .75,  and  .75  by  375,  and  find  the  sum  and 
the  difference  of  the  quotients. 

88.  A  loaded  wagon  weighs  2  T.  3  cwt.  48  lb. ;  the  wagon 
itself  weighs  18  cwt.  75  lb.  The  load  consists  of  215  packages, 
each  of  the  same  weight.  Find  the  weight  of  each,  and  reduce 
it  to  grams  and  kilograms. 

89.  Define  interest,  and  give  and  explain  the  rule  for  com- 
puting the  interest  on  any  sum  of  money,  for  any  time,  and  at 
any  rate  per  cent. 

90.  Extract  the  square  root  of  184.2  to  3  decimals. 

91.  How  many  hektoliters  of  oats  can  be  put  into  a  bin  that 
is  2  meters  long,  1.3  meters  wide,  and  1.5  meters  deep  ? 

Wesleyan  University,  1881. 

92.- Add  t-^  and  if±4i 

I  +  4i  7i  -  4| 

93.  If  money  is  worth  3  per  cent,  what  is  the  premium  on 
government  3^  per  cent  bonds  ? 

94.  How  many  liters  in  6  gallons  of  water? 

95.  How  many  cords  of  stone  will  it  take  to  build  a  Avail 
2  ft.  thick  and  6  ft.  high  about  a  rectangular  cellar  whose  inte- 
rior dimensions,  when  the  wall  is  completed,  shall  be  20  ft.  long 
and  16  ft.  wide? 

96.  How  long  must  a  note  of  $243,  at  ^%,  run  that  its  in- 
terest may  equal  the  int.  on  a  note  of  $125,  for  7  mo.,  at  5^? 

97.  Multiply  \/2  by  a/.123,  and  carry  the  result  to  3  decimals. 

98.  Eeduce  5  mi.  3  fur.  10  rd.  to  kilometers. 

99.  If  5  horses  will  consume  8  bu.  1  pk.  6  qt.  of  oats  in  G  da., 
w^hat  quantity  of  oats  will  7  horses  consume  in  11  da.  ? 


PPENDIX 


Roman    Notation. 

860.  The  Roman  Notation  is  the  method  of  expressing  num- 
bers by  seven  ccqntal  letters,  viz. : 


I, 

V, 

X, 

L, 

0, 

r>, 

M. 

1, 

5, 

10, 

50, 

100, 

500, 

1000. 

861.  To  express  other  numbers,  these  letters  are  combined 
as  in  the  following 

Table. 


I 

II 

III 

IV 

V 

VI 

VII 

VIII 

IX 

X 


1 

2 
3 
4 
5 

6 
7 
8 
9 
10 


XI 

XII 

XIII 

XIV 

XV 

XVI 

XVII 

XVIII 

XIX 

XX 


11 

12 
13 
14 
15 
16 
17 
18 
19 
20 


XXI 

XXV 

XXX 

XXXI 

XL 

L 

LX 

LXX 

XO 

0 


21 
25 

30 
31 
40 

50 
60 
70 
90 
100 


CI  = 

101 

ex  = 

110 

CL  = 

150 

cc  = 

200 

D  = 

500 

DC    rr 

600 

M  = 

1000 

MC  = 

1100 

MD  = 

1500 

MM  = 

2000 

MDCCLXXVI  =  1776.         MDCCCLXXXII  =  1882. 

862.  The  Roman  Notation  is  based  upon  the  following  gen- 
eral principles  : 

1st.  Repeating  a  letter  repeats  its  value.  Thus,  I  denotes 
one ;  II,  two  ;  III,  three ;  X,  ten ;  XX,  twenty,  etc. 

2d.  Placing  a  letter  of  less  value  before  one  of  greater  value, 
dhainislies  the  value  of  the  greater  by  that  of  the  less  ;  placing 
the  less  after  the  greater,  increases  the  value  of  the  greater  by 


360  Appendix. 

that  of  the  less.     Thus,  V  denotes  five,  but  IV  denotes  only 
four,  and  VI  six. 

3d.  Placing  a  horizontal  line  over  a  letter  increases  its  value 
a  thousand  times.  Thus,  I  denotes  a  thousand;  X,  ten  thou- 
sand ;  0,  a  hundred  thousand ;  M,  a  million. 

Notes. — 1.  The  letters  C  and  M  are  the  initials  of  the  Latin  centum,  a 
hundred,  and  mille,  a  thousand. 

2.  The  radix  of  the  system  appears  to  be  doubtful.  Some  have  sup- 
posed that  at  first  it  was  five  (V),  and  %Yas  subsequently  changed  to  ten 
(X),  forming  a  combination  of  the  quinary  and  decimal  systems, 

3.  Others  maintain,  more  plausibly,  that  it  proceeds  according  to  the 
alternate  scale  of  5  and  2,  thus  uniting  the  Unary  with  the  quinary 
scale. 

That  is.     Five  times  one  (I)  are  five  (V). 
Two  times  five  (V)  are  ten  (X), 
Five  times  ten  (X)  are  fifty  (L). 
Two  times  fifty  (L)  are  one  hundred  (C). 
Five  times  one  hundred  (C)  are  five  hundred  (D). 
Two  times  five  hundred  (D)  are  one  thousand  (M). 


English    Numeration. 

863.  By  the  English  Numeration,  numbers  are  divided  into 
periods  oi  six  fgures  each,  and  then  each  period  is  subdivided 
into  units,  tens,  hundreds,  thousands,  tens  of  thousands,  and 
hundreds  of  tliousands,  as  in  the  following 


T 

ABLE. 

5 

S 

5 

ill 

1 

s 

03 

.2 

32 

CD 

o 

o 

Cm 
O 

5M 

s 

o 

o 

o 

O 

i 

!<-( 

3 

00 
O 

O 

s=l 

CS 

3 

X! 

o 

O 

^ 

si 

o 

o 

'*-. 

^ 

.c 

o 

Eh 

J=i 

rjj 

O 

s 

. 

^ 

.£1 

iE 

o 

^ 

5t 

Eh 

.a 

00 

, 

o 

Cm 
O 

rj-j 

Cm 

o 

Cm 

o 

O 

r/1 

tM 

o 

o 

o 

'2 

s 

00 

CO 

'6 

Ol 

s 

73 

QQ 

"■«. 

13 

7J 

s 

13 

■33 

rS 

00 

s 

-a 

00 

*9A 

G 

n 

o 

a 

p 

-«j 

C 

o 

r^ 

c 

••^ 

P 

c 

o 

«—■ 

"» 

S 

a> 

j3 

^ 

<y 

•^ 

S 

s 

.d 

S 

0) 

ki^ 

S 

a; 

J3 

^ 

S 

to 

H 

Eh 

H 

W 

&H 

K 

a 

&H 

H 

W 

Eh 

^ 

til 

Eh 

Eh 

HH 

EH 

4 

0 

7 

6 

9 

2 

9 

5 

8 

6 

0 

4 

4 

1 

3 

0 

5 

6 

V, 

y 

V 

y 

V 

^ 

3d  period.  2d  period.  let  period. 

The  figures  in  the  Table  are  thus  read :  407692  biUions,  958604  millions, 
413  thousand  fifty-six. 


Contractions  in  MultijMcation,  361 


Contractions. 

864.  To  Multiply  any  Number  containing  Two  Figures  by  II. 

The  product  of  any  two  numbers  multiplied  by  11  consists  of 
the  first  figure  of  the  number  multiplied,  the  sum  of  the  two 
figures,  and  the  last  figure. 

Thus,  34x11  11=374,  is  composed  of  3  the  first  figure,  7  the  sum  of 
3  +  4,  and  4  the  last  figure. 

Note.— If  the  sum  of  the  two  ^gure^  exceeds  9,  the  first  or  left-hand 
figure  must  be  increased  by  1, 

1.  What  will  11  tons  of  iron  cost,  at  $45  per  ton  ? 

Analysis. — Since  1  ton  costs  $45,  11  tons  will  cost  11  times  $45  ;  and 
11  times  $45  are  $495. 

2.  Alexander  has  84  marbles,  and  Eichard  has  11  times  as 
many ;  how  many  has  Eichard  ? 

3.  How  many  are  11  times  27  ?  11  times  33  ?  11  times  26  ? 
11  times  34  ?    11  times  62  ?     11  times  54?     11  times  72  ? 

865.  To    Multiply   by   13,   14,  15,  or  I   with    a    Significant   Figure 

annexed. 

If  one  city  lot  costs  13245,  what  will  17  lots  cost  ? 

Analysis. — 17  lots  will  cost  17  times  as  much  as  opekation. 

1  lot.     Placing  the  multiplier  on  the  right,  we  multi-  3245  X  17 

ply  the  multiplicand  by  the  7  units,  set  each  figure  22715 

one  place  to  the  right  of  the  figure  multiplied,  and  — 

add  the  partial  product  to  the  multiplicand.     There-  ^05165,    A  US. 
suit  is  $-)5165. 

866.  To  Multiply  by  21,  31,  41,  etc.,  or  I  with  either  of  the  other 

Significant  Figures  prefixed  to  it. 

If  21  men  can  do  a  job  of  work  in  365  days,  how  long  will  it 
take  1  man  to  do  it? 

OPERATION. 

Explanation. — We  first  multiply  by  the  2  tens,  g/>-        21 

and  set  the  first  product  figure  in  tens'  place  ;  then  ^^ 

adding  this  partial  product  to  the  multiplicand,  we  '  '^ 

have  7S65  for  the  answer.  7665  days,  AllS. 


362  Aijpendix, 

867.  To  Multiply  Two  or  More  Numbers  by  the  Same  Multiplier. 

1.  A  grocer  sold  4  pounds  of  tea  to  one  caslomer,  3  lb.  to 
another,  and  5  lb.  to  another;  how  much  did  it  all  come  to, 
at  7  dimes  a  pound  ? 

Solution.     (4  +  3  +  5)  x  7  =  84  dimes,  Ans.     Hence,  tlie 

^i]!^^.  —  Miiltiplij  the  siuih  of  the  iiin7ibers  by  the  Jiiidti- 
plier. 

868.  To   Multiply  a    Mixed   Number,  whose   Fractional  Part  is  |, 

by  itself. 

1.  What  is  the  product  of  3|^  into  3|^  ? 

Solution. — The  integral  part  of  tlie  given  number  is  3,  and  3  +  1  =4. 
Now  3  into  4  =  12,  and  12  +  ]  =  12],  Ans     Hence,  the 

Rule. — Multiply  the  ijite^ral  part  by  one  more  than 
itself,  and  to  this  result  annex  ^. 

869.  To  Multiply  by  9,  99,  999,  or  any  number  of  9's. 

1.  How  much  will  99  carriages  cost,  at  235  dollars  apiece  ? 

Explanation.  —  Since    1    carriage  opekation. 

costs  $235,  100  carriages  will  cost  100  $23500  Price  of  100  C. 

times  as  much,  or  $23500.      But  99  is  1  qo;^  an       -in 

less  than  100  ;  therefore,  subtracting  tlie         

price  of  1  carriage  from  the  price  of  100  $23265  ^'      ^^     99  C. 
gives  the  price  of  99  carriages. 

870.  To  Divide  by  5. 

1.  A  merchant  laid  out  $873  in  flour,  at  $5  a  barrel ;  how 
many  barrels  did  he  get  ? 

Explanation. — We  first  multiply  the  dividend  " '  "^ 

by  2,  and  then  divide  tlie  product  by  10,  by  cutting  2 

off  the  right-hand  figure.      The  hgure  cut    oli'  is 
written  over  the  divisor,  and  the  fraction,  reouced. 


110  )  17416 


to  its  lowest  terms,  is  annexed  to  the  quotient.  ^^4|,   AnS. 

Note. — This  contraction  depends  upon  the  princij)le  that  any  given  di- 
visor is  contained  in  any  given  dividend  just  as  many  times  as  twice  that 
divisor  is  contained  in  twice  that  dividend,  three  times  that  divisor  in  three 
t'mes  that  dividend,  etc. 


Contractions  in  Division, 


363 


871.  To  Divide  by  25. 

1.  A  farmer  paid   $150   for  cows,  at   $25 
apiece ;  how  many  cows  did  he  buy  ? 

Explanation. — We  first  multiply  the  dividend  by 
4,  and  then  divide  the  product  by  100.     (Art.  118.) 


OPERATION. 

150 
4 


1100  )  6100 
Ans.  6  cows. 


872.  To  Divide  by  125. 

1.  A  man  bought  land  for  $12150,  at  $125  an  acre ;  how 
many  acres  did  he  buy  ? 


Explanation. — We  multiply  the  dividend  by 
8,  and  divide  the  product  by  1000.     (Art.  118.) 

Placing  the  remainder  over  the  divisor,  we  re- 
duce the  fraction  to  lowest  terms,  and  annex  it  to 
the  quotient.     The  answer  is  97^  acres. 

Note. — This  contraction  multiplies  both  the 
dividend  and  divisor  by  8.     (Art.  119,  3°.) 


OPERATION. 

12150 


11000  )  971200 
Ans.  97-iA. 


873.  Demonstration  of  Finding  the  Greatest  Common  Divisor  by 
Continued  Divisions. 


1.  Find  the  g.  c,  d.  of  30  and  42. 


Ans.  6. 


Two  points  are  to  be  proved ; 

1st.  That  6  is  a  common,  divisor  of  the  given  numbers. 

2d.  That  6  is  their  greatest  common  divisor. 

That  6  is  a  common  divisor  of  30  and  42  is  easily  proved  by  trial. 

Next,  we  are  to  prove  that  6  is  the  greatest  common  divisor  of  30  and  42. 
If  the  greatest  common  divisor  of  these  numbers  is  not  6,  it  must  be  either 
greater  or  less  than  6.  But  we  have  shown  that  6  is  a  common  divisor  of 
the  given  numbers  ;  therefore,  no  number  less  than  6  can  be  the  greatest 
common  divisor  of  them.  The  assumed  number  must  therefore  be  greater 
than  6. 

By  the  supposition,  this  assumed  number  is  a  divisor  of  30  and  42  ; 
hence,  it  must  be  a  divisor  of  their  difference,  42  —  30,  or  12.  And  as  it  is 
a  divisor  of  12,  it  must  also  divide  the  product  of  12  into  2,  or  24.  Again, 
since  the  assumed  number  is  a  divisor  of  30  and  24,  it  must  also  be  a  di- 
visor of  their  difference,  which  is  6  ;  that  is,  a  greater  number  will  divide 
a  less  without  a  remainder,  which  is  impossible.  Therefore,  6  must  be 
the  greatest  common  divisor  of  30  and  42. 


364  Appendix. 

874.  To  find  the  Excess  of  9's  in  a  Number. 

1.  Let  it  be  required  to  find  the  excess  of  9's  in  7548467. 

Beginning  at  the  left  hand,  add  the  figures  together,  and  as 
soon  as  the  sum  is  9  or  more,  reject  9  and  add  the  remainder  to 
the  next  figure,  and  so  on. 

Adding  7  to  5,  the  sum  is  12.  Rejecting  9  from  12,  leaves  3  ;  and  3 
added  to  4  are  7,  and  8  are  15.  Rejecting  9  from  15,  leaves  6  ;  and  6  added 
to  4  are  10.  Rejecting  9  from  10,  leaves  1 ;  and  1  added  to  6  are  7,  and  7 
are  14.     Finally,  rejecting  9  from  14  leaves  5,  the  excess  required. 

875.  Hence  we  derive  this  property  of  the  number  9 : 

Any  nurnher  divided  hy  9  will  leave  the  same  remainder  as 
the  sum  of  its  digits  divided  hy  9. 

Notes. — 1.  It  will  be  observed  that  the  excess  of  9's  in  any  tico  digits  is 
always  equal  to  the  sum,  or  the  excess  in  tlie  sum,  of  those  digits.  Thus, 
in  15  the  excess  is  6,  and  1  +  5  =  6;  so  in  51  it  is  6,  and  5  +  1  =  6. 

2.  The  operation  of  finding  the  excess  of  9's  in  a  number  is  called  cast- 
ing out  the  9's. 

2.  What  is  the  product  of  746  multiplied  by  475  ? 

OPERATION.  Proof  by  Excess  of  G's.  Proof  by  Mult. 

746  Excess  of  9's  in  multiplic'd  is  8  475 

475  "            "       multiplier  is    7  746 

3730       Now 8x7  =  56  2850 

5222  '^^16  excess  of  9's  in  56  is         2  iqqo 

2984  The  excess  of  9's  in  product  is  2  3325 


Ans.  354350        ^^^ ^-^      Ans.  354350 

876.  To  prove  Multiplication  by  Excess  of  9's. 

Fijul  the  excess  of  9's  in  each  factor  separately ;  then 
multiply  these  excesses  together,  and  reject  the  9's  from 
the  result ;  if  this  excess  agrees  with  the  excess  of  9's  in 
the  answer,  the  worh  is  right. 

Note. — The  preceding  is  not  a  necessary  but  an  incidental  property  of 
the  number  9.  It  arises  from  the  hno  of  increase  in  the  decimal  notation. 
If  the  rndix  of  the  system  were  8,  it  would  belong  to  7  ;  if  the  radix  were 
12,  it  would  belong  to  11  ;  and  universally,  it  belongs  to  the  number  that 
is  one  less  than  the  radix  of  the  svpter.i  of  notation. 


Circulating  Decimals.  365 


Circulating    Decimals. 

877.  A  Circulating  Decimal  is  one  in  which  the  same  figure 
or  set  of  figures  is  continually  repeated  in  the  same  order. 

878.  In  reducing  common  fractions  to  decimals,  we  find  that 
in  one  class  of  examples  the  division  is  complete,  and  the  quo- 
tient is  an  exact  decimal. 

In  another  class,  the  same  figure  or  set  of  figures  is  repeated 
again  and  again,  and  the  division  will  not  terminate,  though 
continued  indefinitely. 

The  former  are  Terminate  Decimals,  the  latter  are  Circulating 
Decimals,  and  the  figure  or  figures  repeated  the  Repetend. 

Thus,  4  =  .5,  f  =  .4,  I  =  .75,  |  =  .625,  etc.,  are  exact  decimals. 
But  i  =  .333333  +  ,  if  =  .424242  +  ,  i|  ^  .297297  +  ,  are  interminate. 

879.  By  inspection  we  see  that  the  denominators  of  the  first 
class  are  t\iQ  prime  numbers  2  or  5,  or  are  composed  of  the  fac- 
tors 2  or  5. 

In  the  second  class,  the  denominators  contain  other  prime 
factors  than  2  and  5. 

880.  To  find  whether  a  common  fraction  will  produce  a  ter- 
minate or  a  circulating  decimal, 

Resolve  the  cleiioiiiiuator  into  its  prime  factors.  If  it 
contains  no  other  factors  than  2  and  5,  the  quotient  will 
he  a  terminate  decimal. 

If  it  contains  any  other  prime  factors  than  2  and  5, 
the  quotient  will  he  a  circulating  decimal. 

881.  Circulating  decimals  are  expressed  by  writing  the  rep- 
etend once,  and  placing  a  dot  over  the^r^^  and  last  figure. 

Thus,  the  repetends  .33333+  and  .297297+  are  written  .3  and  .297. 

882.  When  the  repetend  begins  with  tenths,  the  decimal  is 
called  a  Pure  Repetend ;  as,  .4 ;  .297. 


1. 

|. 

5. 

-AV- 

9. 

2. 

i- 

6. 

1  3 
16* 

10. 

3. 

A- 

7. 

7 
16* 

11. 

4. 

e- 

8. 

4. 

12. 

36G  Appendix. 

883.  When  the  repetend  is  preceded  by  one  or  more  decimal 
figures,  it  is  called  a  Mixed  Eepetend;  as  .27;   .42631. 

Note. — The  decimal  figures  hefore  the  repetend  are  called  the  finite 
part  of  the  decimal ;  as  2  and  43  in  the  mixed  repetends  above. 

884.  Change  the  following  fractions  to  terminate  or  circu- 
latimj  decimals,  and  mark  the  repetends  in  each.      (Art.  249.) 

JLj6  14      1_6  18      _3JL 

21-  •^*'     30'  ■'■°*     180* 

_9  15       2.i  iQ      2l1. 

-T-  16       2_2  on      _3 


885.  To  Reduce  a  Pure  Repetend  to  a  Common  Fraction. 

1.  Reduce  .24  to  a  common  fraction. 

Analysis. — Since  the  repetend  con.  operation. 

sists  of  two  figures,  if  we  multiply  it  by  q  \\   ^  i  qq  94  2424 

100,  the  decimal  part  of  the  product  mil 

be  the  same  as  the   repetend.     Now,  if  0.24  X  1        =     0.2424 

we  subtract  the  repetend  from  this  prod-         ^  a  ; 

0  24  y  99     —  24 
uct,  the  remainder  will  have  no  decimal  y^  'J^j  '^■^ 

and  will  be  99  times  the  given  repetend.  ^24  :zr  ^  =  -S_    Ans, 

Therefore,  once  the  given  repetend  is  f  f 
or  3^,  Ans.    Hence,  the 

EuLE. —  Take  the  repetend  for  the  numerator,  and 
Tnahe  the  denominator  as  many  9's  as  there  are  figures 
in  the  repetend.  Reduce  the  fraction  thus  produced  to 
its  lowest  terms. 

Reduce  the  following  to  common  fractions: 


2. 

.18. 

5. 

.123. 

8. 

.1007. 

11. 

.25121. 

3. 

.72. 

6. 

.297. 

9. 

.6435. 

12. 

.142857. 

4. 

.69, 

7. 

.045. 

10. 

.4158. 

13. 

.076923. 

Circidating  Dechnals,  367 

886.  To  Reduce  a  Mixed  Repetend  to  a  Common  Fraction. 

14.  Keduce  0.227  to  a  common  fraction. 

OPERATION. 

SoLUTioN.-Subtracting  the  finite  part  ^27  Given  decimal, 

from  the  given  mixed  repetend,  both  re-  t?-    • 

garded  as  integers,  we  have  for  the  uumer- ^^^^^^^  paiT. 

ator  235,   and   for   the   denominator  990,  ^^5  Numerator, 

that  is,  as  many  9's  as  there  are  figures  in  990  Denominator. 

the   repetend   with   as  many   ciphers  an-  22^5  4_5    .5       i^<. 

nexed  as  there  are  figures  m  the  jimte  part, 

and  the  result  is'fff  =  jfg  =  A,  Ans.     Hence,  the 

EuLE. — For  the  numerator,  siibtractthe  finite  part  from 
the  given  mijoecl  repetend,  both  regarded  as  integers ; 
and  for  the  denominator,  talce  as  many  9's  as  there  are 
figures  in  the  repetend,  with  as  many  ciphers  annexed 
as  there  are  figures  in  the  finite  part. 

Explanation. — Since  the  repetend  operation. 

has  two  figures,   if  we  multiply  the  q  927  x  100  =  22.7272 
given  mixed  repetend  by  100,  and  from 

the  product   subtract   once   the  given  0.227  X  1        =        .2272 

mixed  repetend,  the  remainder  (22.5)  •  ;,  

will  be  99  times  the  given  mixed  repe-  ^•'^^^    X  J9     —  22.5 

tend;  and  once  the  mixed  repetend  =  0.227  ^^   ^^"^   rr:  ^^^ 

^^^  =  ||3.     But  225  is  the  difierence 

between  327  the  given  mixed  repetend,  TWS  '^^  TtE  ^^^  2T?  -^  '^^* 

regarded  as  an  integer,  and  2  the  finite 

part  of  it,  regarded  as  an  integer,  and  f |f  =  ^%,  the  same  as  before. 

Change  the  following  to  common  fractions : 

15.  .6472.         17.   .5925.         19.   .5925.         21.  .9285714. 

16.  .1004.         18.   .0227.         20.   .4745.         22.   .008497133. 

887.  Circulating  decimals,  when  reduced  to  common  frac- 
tions, may  be  added,  subtracted,  multiplied,  and  di\dded,  like 
other  common  fractions. 

23.  What  is  the  sum  of  .5925  +  .4745  +  .0227  ? 

24.  What  is  the  difference  between  .6435  and  ,4158  ? 


368 


Appendix. 


Table  of  Prime  Numbers  from  I  to  1009. 


1 

59 

139 

233 

337 

439 

557 

653 

769 

883 

2 

61 

149 

239 

347 

443 

563 

659 

773 

887 

3 

67 

151 

241 

349 

449 

569 

661 

787 

907 

5 

71 

157 

251 

353 

457 

571 

673 

797 

911 

7 

73 

163 

257 

359 

461 

577 

677 

809 

919 

11 

79 

167 

263 

367 

463 

587 

683 

811 

929 

13 

83 

173 

269 

373 

467 

593 

691 

821 

937 

17 

89 

179 

271 

379 

479 

599 

701 

823 

941 

19 

97 

181 

277 

383 

487 

601 

709 

827 

947 

23 

101 

191 

281 

389 

491 

607 

719 

829 

953 

29 

103 

193 

283 

397 

499 

613 

727 

839 

967 

31 

107 

197 

293 

401 

503 

617 

733  ^ 

853 

971 

37 

109 

199 

307 

409 

509 

619 

739 

857 

977 

41 

113 

211 

311 

419 

521 

631 

743 

859 

983 

43 

127 

223 

313 

421 

523 

641 

751 

863 

991 

47 

131 

227 

317 

431 

541 

643 

757 

877 

997 

53 

137 

229 

331 

433 

547 

647 

761 

881 

1009 

SURVEYOR'S    Measure. 

888.  Surveyor's  Measure  is  used  in  measuring  land,  in  lay- 
ing ont  roads,  establishing  boundaries,  etc. 

889.  The  Linear  Unit  usually  employed  by  surveyors  is 
Guntefs  Chain,  which  is  4  rods  or  i!>Q  feet  long,  and  contains 
100  links.     It  is  subdivided  as  in  the  following 


Ta  b  l  e. 

7.92  iuclies  {in.)  =  1  link, I, 

25  links  =  1  rod  or  pole,     .     .  r. 

4  rods  =  1  cJuiin,      .     .     .  ch. 

80  cliains  =  1  mile, m. 

Notes. — 1.   Ountefs  chain  is  so  called  from  the  name  of  its  inventor. 
Measurements  by  it  are  usually  given  in  chains  iir\6.  hundredths  of  a  chain. 

2.  In  measuring  roads,  etc.,  engineers  use  a  chain,  or  measuring  tape, 
100  feet  long,  each  foot  being  divided  into  tenths  and  hundredths. 

3.  The  mile  of  the  table  is  the  common  land  mile,  which  contains  5380 
feet. 


Government  Lands.  369 

890.  The  Measuring  Unit  of  Land  is  the  Acre. 

Table. 

625  sq.  links  =   1  sq,  rod  or  pole,  .     sq,  rd. 

16  sq.  rods  =  1  sq.  chain,      .     .  .     sq.  c. 

10  sq.  chains,  or  )  ^                                         . 
160  sq.  rods            ) 

640  acres  =  1  sq.  mile,  .     .     .  .     sq.  mi. 

Notes. — 1.  The  Rood  of  40  square  rods  is  no  longer  a  unit  of  measure. 
2.  A  Square,  in  Architecture,  is  100  square  feet. 

G-OVERNMENT     LaNDS. 

891.  The  i^iiblic  lands  of  the  United  States  are  divided  into 
Townships,  which  are  subdivided  into  Sections,  Half-Sections, 
Quarter-Sections,  etc. 

A  Towiifship  is  6  miles  square,  and  contains  36  sq.  mUes. 

A  Section  is  1  mile  square,  and  contains  640  acres. 

A  Half-Section  is  1  mile  long  by  ^  mile  wide,  and  contains  320  acres. 

A  Quarter-Section  is  160  rods  square,  and  contains  160  acres. 

892.  The  method  adopted  by  the  GoYernment  in  siirvejring 
a  new  territory  is  the  following  : 

First. — A  line  is  run  North  and  South,  called  the  Principal 
Meridian. 

Second. — A  line  is  run  on  a  parallel  of  latitude  E.  and  W. 
called  the  Base  Line. 

TJiird. — Lines  are  run  6  miles  apart  parallel  to  the  principal 
meridian. 

Fourth. — Other  lines  are  run  6  miles  apart  parallel  to  the 
base  line,  forming  townships,  or  squares^  each  containing  36  sq.    • 
miles. 

893.  Townships  are  designated  by  their  number  N.  or  S.  of 
the  base  line. 

894.  A  line  of  townships  running  N".  and  S.  is  called  a 
Range,  and  is  designated  by  its  number  E.  or  W.  of  the  prin- 
cipal meridian.     Thus, 


370 


Appendix. 


T.  39  N.,  R.  14  E.  3d  P.  M.,  describes  the  township  ihat  is  in  the  39th 
tier  North  of  the  base  line,  and  in  the  14th  range  E.  of  the  3d  principal 
meridian. 

895.  A  Township  is  divided  into  Sections  each  1  mile  square 
and  contains  640  acres.     Thus, 


A  Section 

=  1  mi.   X 

1  mi.  —  640  acres 

A  Half  Section 

=    1     "         X 

i    "     =  320      " 

A  Quarter  Section 

=     1     "         X 

i    "     =  160     " 

A  Hall-quarter  Section 

=     1     "         X 

i    "     =     80     " 

A  Quarter-quarter  Section 

=     1     "         X 

tV  "     =     40      " 

6 

5 

4 

3 

2 

1 

7 

8 

9 

10 

11 

13 

18 

17 

16 

15 

14 

13 

19 

20 

21 

23 

23 

24 

30 

29 

28 

27 

26 

25 

31 

32 

33 

34 

35 

36 

The  adjoining  diagram  represents  a  A  Township. 

township  divided  into  sections,  which  N 

are  numbered  commencing-  at  the  N.E. 
corner,  and  running  W.  in  the  North 
tier,  E.  in  the  second,  etc. 

Each  section  is  divided  into  4  quar- 
ter sections,  called  N.E.,  S.E.,  N.W., 
and    S.W.    quarters,   each   containing     w 
160  acres. 

Thus,  S.E.  \,  sec.  16,  T.  39  N.,  R. 
14  E.  3d.,  P.  M.,  is  read,  "Southeast 
quarter  of  section  16,  tier  39  north, 
range  14  east  of  third  principal  meri- 
dian." 

1.  A  colony  of  224  persons  took  \\])  a  township  of  land  and 
divided  it  equally  among  them ;  how  many  acres  did  each 
receive  ? 

2.  What  part  of  a  section  did  each  colonist  receive,  and  what 
did  it  cost  him,  at  11.25  an  acre  ? 

3.  What  will  it  cost  to  enclose  a  quarter  section  of  land  with 
a  fence  5  rails  high,  at  $2  for  every  3  rods  ? 

4.  If  you  pay  $1.75  an  acre  for  a  half  section  of  land,  and 
sell  a  quarter  section  for  12.50,  how  much  will  your  remaining 
quarter  cost  you  ? 

5.  A  company  of  speculators  bought  a  township  at  11.50  an 
acre ;  they  sold  10  sections  at  12.25  an  acre,  15  sections  at 
$3.50,  8  sections  at  14,  and  the  balance  at  S5  an  acre ;  how 
much  did  they  sell  at  15,  and  what  was  the  gain  on  the  whole  ? 
Explain  by  diagram. 


Pounds  in  a  Bushel. 


371 


896.   Table  of  Pounds  Avoirdupois  in  a  Bushel,  as  fixed   by  Law 
in  the  several  States  named. 

It  is  becoming  common  in  some  parts  of  this  country  and  in 
England  to  sell  grain  and  other  produce  by  weight  and  not  by 
measure,  a  much  more  equitable  system  than  that  which  has 
long  prevailed. 


w 

K 
•<s> 

!S 

b-l 

COMMODITIES. 

60 
52 

no 

o 
56 

60 
56 

1 

60 
52 

-^ 

60 
56 

60 
50 

5J> 

« 

60 
56 

1 

60 
56 

« 

1 

60 
56 

60 
56 

o 

1 
60 

60 

1 

S 

60 

58 

60 
54 

•i 

60 
56 

1 

60 
56 

"Si 

60 
56 

o 

60 
56 

s 

1 

60 
56 

i 

60 
56 

Wheat 

Indian  Com  in  ear 

52  56 

Oats     

32 

50 
40 

28 
45 

32 

48 
40 

32 

48 
50 

35 
48 
52 

48 
52 

32 
32 

30 
46 
46 

32 

48 
42 

32 
48 
42 

35  30 
48  4R 

32 

48 

48 

48 
50 

32 

48 

34 
46 
42 

32 
47 

48 

32 
46 
46 

36 
45 
42 

32 
48 
42 

Barlev  .... 

Buckwheat 

52 

50 

Rve 

54 

56 

54 
60 

56 
60 

56 
60 

56 
60 

32 

56 

56 
60 

56 
60 

56 
60 

56 

56 
60 

56 
60 

56 
60 

56 

56 

56 
60 

56 

60 

Clover  Seed  

Timothy  Seed  — 

45 

45 

45 

45 

45 

45 

46 

Blue  Grass  Seed. . 

14 

14 

14 

14 

14 

56 

Flax  Seed 

56 

56 

56 

56 

56 

55 

55 

56 

Hemp  Seed 

4i 

44 

44 

44 

44 

Notes. — 1.  Beans,  peas,  and  potatoes  are  usually  estimated  at  60  lb.  to 
the  bii.,  but  the  laws  of  N.  Y.  make  63  lb.  of  beans  to  a  bushel. 

In  Illinois,  50  lb.  of  common  salt,  or  55  lb,  fine,  are  1  bu.  In  N.  J., 
56  lb.  of  salt  are  1  bu.  In  Ind.,  Ky.,  and  Iowa,  50  lb.  are  1  bu.  In  Penn., 
80  lb.  coarse,  70  lb.  ground,  or  63  lb.  fine  salt  are  1  bu. 

In  Maine,  30  lb.  oats,  and  64  lb.  of  beets  or  of  ruta-baga  turnips  are  1  bu. 

In  New  Hampshire,  30  lb.  of  oats  are  1  bu. 

3.  Grain,  seeds,  and  small  fruit  are  sold  by  the  bushel,  stricken  or  level 
measure. 

Large  fruit,  potatoes,  and  all  coarse  vegetables  by  heaped  measure. 


897.    Capacity  Measures,  estimated  by  Avoirdupois  Weight. 


63|  pounds,  or  1000  oz. 

100  pounds 

100  pounds 

196  pounds 

200  pounds 

280  pounds 

§40  pounds 


1  cubic  foot  of  water. 

1  keg  of  nails. 

1  quintal  of  dry  fish. 

1  barrel  of  flour. 

1  barrel  offish,  beef,  or  porko 

1  barrel  of  salt. 

1  cask  of  lime. 


372  Appendix, 


Apothecaries'    Fluid    Measure. 

898.  Apothecaries'  Fluid  Measure  is  used  in  mixing  liquid 
medicines. 

60  minims,  or  drops  (Tt|^  ot  gtt.)  =  1  fluid  draclim,     .    .  f  3  . 

8  fluid  draclims  —  1  fluid  ounce,  .     .    .  /s  . 

16  fluid  ounces  =  1  pint 0. 

8  pints  =  1  gallon, .....  Cong. 

Note. — Gtt.  for  guttce,  Lsitm,  signifying-  drops;  0,  for  oaarius,  Latin 
for  one-eighth ;  and  (Jong.y  congiariwm,  Latin  for  gallon. 

899.  The    following    approximate   measures,   though    not 
strictly  accurate,  are  often  useful  in  practical  life: 

45  drops  of  water,  or  a  common  teaspoonful  =  1  fluid  drachm. 

A  common  tablespoonful  =   |  fluid  ounce. 

A  small  teacupful,  or  1  gill  =  4  fluid  ounces. 

A  pint  of  pure  water  =  1  pound. 

4  tablespoonfuls,  or  a  wine-glass  =  \  gill. 

A  common -sized  tumbler  =  4  pint. 

4  teaspoonfuls  =  1  tablespoonful. 


900.   The  following  linear  units  are  often  used : 

11  statute  miles  =  1  geographic  or  nautical  mile. 

60  geographic,  or  )         a  a  *i  * 

«^.  .  ,     -  =  1  degree  on  the  equator. 

69|  statute  mi.,  nearly, ) 

360  degrees  =  1  circumference  of  the  earth. 

A  knot,  used  for  measuring  distances  at  sea,  is  equivalent  to  a  nautica 
DiUe. 

4  inches      =  1  hand,  for  measuring  the  height  of  horses. 
9  inches      =  1  span. 

18  inches  =  1  cubit. 

6  feet  =  1  fathom,  for  measuring  depths  at  sea, 

120  fathoms  =  1  cable's  length. 

3.3  feet  =  1  pace,  for  measuring  approximate  distances. 

5  paces  =  1  rod,  "  "  " 


Annual  Interest  373 


Leap    Years. 

901.  A  Solar  Day  is  the  time  between  the  departure  of  the 
sun  from  a  given  meridian  and  his  return  to  it. 

902.  A  Mean  Solar  Day  is  the  average  lengtli  of  all  the 
solar  days  in  the  year,  and  is  divided  into  24  hours,  the  first  12 
being  designated  by  a.  m.,  the  last  by  p.  m. 

Note. — a.m.  is  au  abbreviation  of  ante  meridies,  before  midday ;  p.  m., 
of  post  meridies,  after  midday. 

903.  A  Solar  Year  is  the  time  in  which  the  earth,  starting 
from  one  of  the  tropics  or  equinoctial  points,  revolves  around 
the  sun,  and  returns  to  the  same  point.  It  is  thence  called  the 
tropical  year,  aud  is  equal  to  365  da.  5  hr.  48  min.  49.7  sec. 

Notes. — 1.  The  excess  of  tlie  solar  above  the  common  year  is  6  hours 
or  i  of  a  day,  nearly ;  hence,  in  4  years  it  amounts  to  about  1  day.  To 
provide  for  this  excess,  1  day  is  added  to  the  month  of  February  every  4th 
year,  which  is  called  Leap  year,  because  it  leaps  over  the  limit,  or  runs  on 
1  day  more  than  a  common  year. 

2.  Every  year  that  is  exactly  divisible  by  4,  except  centennial  years,  is 
a  leap  year ;  the  others  are  common  years.  Thus,  1876,  '80,  etc.,  were 
leap  years';  1879,  '81,  were  common.  Every  centennial  year  exactly  di- 
visible by  400  is  a  leap>  year;  the  other  centennial  years  are  common. 
Thus,  1600  and  2000  are  leap  years  ;  1700,  1800,  and  1900  are  common. 

Annual    Interest. 

904.  Annual  Interest  is  interest  that  is  payable  every  year. 

905.  To  Compute  Annual  Interest ^  when  the  Principal,  Rate, 

and  Time  are  given. 

1.  What  is  the  amount  due  on  a  note  of  $500,  at  6^,  in  3  yr. 
with  interest  payable  annually  ? 

SOLUTION. 

Principal $500.00 

Interest  for  1  year  is  $30  ;  for  3  years  it  is  $30  x  3,  or 90.00 

Interest  on  1st  annual  interest  for  2  yr.  is 3.60 

2d        "            "         "    l''-    is 1.80 

The  amount  is $595.40 


374  Appendix. 

Rule. — Find  the  interest  on  the  pTincipal  for  the  given 
time  and  rate ;  also  find  the  simple  legal  int.  on  eaeh 
annual  int.  for  the  time  it  has  remained  unpaid. 

The  sum  of  the  principal  and  its  int.,  with  the  int.  on 
the  unpaid  annual  interests,  will  he  the  amount. 

Note. — When  cotes  are  made  payable  "with  interest  annually,"  sim- 
ple interest  can  be  collected,  in  most  of  the  States,  on  the  annual  interest 
after  it  becomes  due.  This  is  according  to  the  contract,  and  is  an  act  of 
justice  to  the  creditor,  to  comjjensate  him  for  the  damage  he  suffers  by  not 
receiving  his  money  when  due. 

2.  What  is  the  amount  of  a  note  of  $1500,  payable  in  4  yr. 
3  mo.  10  da.,  with  int.  annually  at  5%? 

906.   Connecticut  Rule  for  Partial  Payments. 

I.  When  the  first  payment  is  a  year  or  more  from  the  time 
the  interest  commenced : 

Find  the  amount  of  the  principal  to  that  time.  If  the 
payment  equals  or  exceeds  the  interest  due,  subtract  it 
from  the  amount  thus  found,  and  considering  the  re- 
mainder a  new  principal,  proceed  as  before. 

II.  When  a  payment  is  made  before  a  year's  interest  has 
accrued : 

Find  the  amount  of  the  principal  for  1  year ;  also,  if 
the  payment  equals  or  exceeds  the  interest  due,  find,  its 
amount  from  the  time  it  was  made  to  the  end  of  the 
year ;  then  subtract  this  a?nount  from  the  amount  of  the 
principal,  and  treat  the  remainder  as  a  nciu  principal. 

III.  If  the  payment  be  less  than  the  interest : 

Subtract  the  payment  only  from  the  amount  of  the 
principal  thus  found,  and  proceed  as  before. 


$650.  New  Havei^,  April  12,  1878. 

1.  On  demand,  I  promise  to  pay  to  the  order  of  George  Sel- 
den,  six  hundred  fifty  dollars,  with  interest,  value  received. 

Thomas  Sawyer. 

Indorsements:— May  1,  1879,  rec'd  $116.20.  Feb.  10,  1880, 
rec'd  $61.50.  Dec.  12, 1880,  rec'd  $12.10.  June  20, 1881,  rec'd 
$110.     What  was  due  Oct.  21,  1881  ? 


Partial  Paijments.  375 

SOLUTION. 

Principal,  dated  April  12,  1878 $650.00 

Interest  to  first  payment,  May  1, 1879  (1  yr.  19  da.) 41.06 

Amoimt,  May  1,  '79 691 .06 

First  payment,  May  1,  79 116.20 

Remainder,  or  New  Principal,  May  1,  '79 574.86 

Interest  to  May  1,  '80,  or  1  yr.  (2d  payment  being  short  of  1  yr.). .       34.49 

Amount,  May  1,  '80 609.35 

Amount  of  second  payment  to  May  1,  '80  (2  mo.  20  da.) 62.32 

Remainder,  or  New  Principal,  May  1,  '80 547.03 

Amount,  May  1,  '81  (1  yr.) 579.86 

Third  payment  (being  less  than  interest  due)  draws  no  interest. . .       12.10 

Remainder,  or  New  Principal,  May  1,  '81 567.76 

Amount,  Oct.  21,  '81  (5  mo.  20  da.), .    583.85 

Amount  of  last  payment  to  settlement  (4  mo.  1  da.) 112.22 

Balance  due  Oct.  21,  '81 $471.63 

Note. — For  additional  exercises  in  the  Connecticut  Rule,  the  student  is 
referred  to  Art.  554. 


907.   Vermont    Rule   for    Partial    Payments    on    Notes    bearing 

Annual  Interest. 

I.  When  payments  are  made  on  notes  bearing  interest,  such 
payments  shall  be  applied, 

"  First,  to  liquidate  the  interest  that  has  accrued  at  the 
time  of  such  payments ;  and  secondly,  to  the  extinguish- 
ment of  the  principal. " 

II.  When  notes  are  made  ^^  with  interest  annually," 

Tlie  annual  interests  which  remain  unpaid  shall  he 
subject  to  simple  interest  froin  the  time  they  become  due 
to  the  time  of  settlement. 

III.  If  payments  have  been  made  in  any  year,  reckoning 
from  the  time  such  annual  interest  began  to  accrue,  the  amount 
of  such  payments  at  the  end  of  such  year,  with  interest  thereon 
from  the  time  of  payment,  shall  be  applied : 


376  Appendix. 

"  First,  to  liquidate  the  simple  interest  that  has  accrued 
from  the  unpaid  annual  interests. 

"  Secondly,  To  liquidate  the  annual  interests  that  have 
hecome  due. 

"  Thirdly,  To  the  extinguishment  of  the  principal. 


^1500.  BuRLiKGTOJS",  Fel.  1,  1877. 

1.  On  demand,  I  promise  to  pay  to  tlie  order  of  Jared  Sparks, 
fifteen  hundred  dollars,  with  interest  annually  at  6^,  value  re- 
ceived. Augustus  Wakren. 

Indorsements :— Aug.  1,  1877,  received  1160;  Nov.  1,  1880, 
$250.     Kequired  the  amount  due  Feb.  1,  1881. 

SOLUTION. 

Principal $1500.00 

Annual  interest  to  Feb.  1,  78  (1  yr.  at  6;^ 90.00 

Amount 1590.00 

First  payment,  Aug.  1,  77 $160.00 

Interest  on  same  to  Feb.  1,  78  (6  mos.) 4.80      104.80 

Remainder,  or  New  Principal 1425.20 

Annual  interest  on  same  from  Feb.  1,  78,  to  Feb.  1,  '81  (3  yr.). . .  256.53 
Interest  on  first  annual  interest  from  Feb.  1,  79  (2  yr.)..     $10.26 

Interest  on  second  annual  int.  from  Feb.  1,  '80  (1  yr.) 5.13  15.39 

Amount 1G97.13 

Second  payment,  Nov.  1,  '80 $250.00 

Interest  on  same  to  Feb.  1,  '81  (3  mo.) 3.75      253.75 

Balance  due  Feb.  1,  '81 $1443.37 

908.    New  Hampshire  Rule  for  Partial  Payments. 

I.   When  on  notes  drawing  annual  interest. 

Find  the  interest  upon  the  ])rincipal  from  date  of  note 
to  the  end  of  the  year  next  after  the  first  payment,  also 
upon  each  annual  interest  to  the  same  date. 

IT.  If  the  first  payt.  be  larger  than  the  sum  of  interests  due, 

Find  the  int.  on  such  payt.  from  the  time  it  was  made 
to  end  of  the  year,  and  deduct  the  sum  of  payt.  a,nd  int, 
from  the  amount  of  principal  and  interests. 


Partial  Payments.  377 

III.  If  less  than  the  annual  interests  accruing, 

Deduct  the  payment  juitJiout  interest  froTn  the  sinn  of 
annual  and  simple  interest,  and  upon  the  balance  of 
such  interest  cast  the  simple  interest  to  the  time  of  the 
next  payment. 

IV.  If  less  than  the  simple  interest  due, 

Deduct  it  from  the  simple  interest,  and  add  the  bal- 
ance witJiout  interest  to  the  other  interests  due  when  the 
next  payment  is  made. 

Proceed  thus  to  the  end  of  the  year  after  the  last  pay- 
ment, being  careful  to  carry  forward  cdl  interest  unpaid 
at  the  end  of  each  year.* 

1.  A  agrees  to  pay  B  $2000  in  6  yr.  from  Jan.  1,  1870,  with 
interest  annually.  On  July  1,  1872,  a  payment  of  1500  was 
made;  and  Oct.  1,  1873,  $50.     What  was  due  Jan.  1,  1876? 

SOLUTION. 

Principal $2000.00 

First  year's  interest ...   $120.00 

3  yr.  simple  int.  thereon 14.40       134.40 

Second  year's  interest 120.00 

1  yr.  simple  int.  thereon 7.20       127.20 

Third  year's  interest 120.00 

$2381.60 

First  payment,  July  1,  1872 $500.00 

Int.  thereon  from  July  1,  '72,  to  Jan.  1,  '73 15.00      515.00 

Balance  of  principal $1866.60 

Interest  on  same  for  fou.rth  year 111.99  + 

Second  payt.  (less  than  the  int.  accruing  during  the  year) 50.00 

Balance  of  fourth  year's  interest  unpaid $61.99  + 

Annual  interest  on  balance  of  principal  for  fifth  year 111.99  + 

"   sixth  "     111.99  + 

Simple  int.  on  unpaid  bal.  of  fourth  year's  int.  for  2  yr 7.43  + 

Simple  interest  on  fifth  year's  interest  for  one  year 6.71  -f- 

Balance  of  principal  ....    1866.60 

Amount  due  January  1,  1876 $2166.71 

*  Abstract  of  N.  H,  Court  Rule,  Report  of  Hon.  C.  A.  Downs,  State 
Superintendent. 


378  Appendix. 

909.  The  Twelve  Per  Cent  Method  of  Computing  Interest. 

1.  Find  the  int.  of  1275.20,  for  3  yr.  4  mo.  10  cla.,  at  12^. 

SOLUTION. 

Int.  of  $275.20, 1  yr.  at  1%  =  $275.20  x  .01  =  $2,752. 
1  yr.  at  12%  =  $2,752  x  12  =  $33,024. 
1  mo.  at  12%  -  12  mo.  at  1%  =  $2,752.     (Art.  578.) 

3  yr.  at  12%  =  $33.024x3 $99,072 

4  mo.  (1  of  1  yr.)  =  $33.024-=-3 11.008 

10  da.  (i  of  mo.)  =  2.752h-3 .917 

Hence,  the  Ans.  $110,997 

EuLE. — For  1  year:  Find  the  interest  on  the  principal 
at  1%,  by  moving  the  decimal  point  two  places  to  the  left, 
and  multiplying  the  result  hy  12. 

For  2  or  more  years :  Multiply  the  interest  for  1  year  hy 
the  nujnber  of  years. 

For  months  and  days :    Proceed  as  in  Art.  537. 

AVERAQE      OF     MIXTURES. 

910.  To  find  the  Average  Value  of  a  Mixture,  when  the  Quan- 

tity and  Price  of  each  Article  are  given. 

1.  A  man  mixed  45  bii.  oats  worth  25  cts.  a  bushel  with 
38  bu.  corn  at  50  cts.,  and  56  bu.  rye  at  60  cts. ;  what  was  the 
mixture  worth  a  bushel  ? 

OPERATION. 

Solution. — The  whole  number  of  bush-  |0. 25  X  45  =  111.  25 

els  mixed  is  45  +  38  +  56  =  139.     The  whole  0  ^0  v  '^K  1  Q  00 

cost  of    the  mixture  is    $11.25  +  $19.00  ^-^^X^O  —     iJ.UU 

+  $33.60  =  $63.85.  0. 60  X  56  =     33.60 

Now,  $63.85-139  =  $0.46  nearly,   the  I39  )  $63.85 
price  of  1  bushel  of  the  mixture.     Hence, 
the 


Ans.  $0.46. 


Rule. — Divide  the  value  of  the  whole  mixture  hy  the 
sum  of  the  articles  mixed. 

Notes. — 1.  If  an  article  costs  nothinpf,  as  water,  its  value  is  0  ;  but  the 
quantity  used  must  be  added  to  the  other  articles. 

2.  The  process  of  finding  the  average  value  of  mixtures  is  often  called 
Allio-ation. 


Average  of  Mixtures. 


379 


2.  A  grocer  had  three  kinds  of  sugar,  worth  6,  8,  and  12 
cents  per  pound;  he  mixed  112  lb.  of  the  first,  150  lb.  of  the 
second,  and  175  of  the  third  together.  What  was  the  mixture 
worth  per  pound  ? 


5s. 

JL 
4 

8s. 

1 

lis. 

i 

12s. 

\ 

911.  To  find  the  Propovtiotial  Parts  of  a  Mixture,  the  Mean 
Price  and  the  Price  of  each  Article  being  given. 

3.  A  grocer  desired  to  mix  4  kinds  of  tea,  worth  5s.,  8s., 
lis.,  and  12s.  a  pound,  so  that  the  mixture  should  be  worth  9s. 
a  pound  ;  in  what  proportion  must  they  be  taken  ? 

Analysis. — First  find  how  much  it  takes  operation. 

of  each  article  to  gain  or  lose  a  w;< /if  of  the  QqJ    123 

mean  price.  Since  the  mean  price  is  9s.  a  — 
pound,  1  lb.  at  5s.  gains  4s,  ;  hence,  to  gain 
Is.  takes  \  lb,,  which  we  place  in  Col.  1. 
Again,  1  lb.  at  12s,  loses  3s.  ;  hence,  to  lose  9s. 
Is.  takes  |  lb.,  which  we  place  also  in  Col.  1, 
opposite  the  price  compared.  In  like  man- 
ner, 1  lb.  at  8s.  is  required  to  gain  Is.,  while 

1  lb.  at  lis,  loses  2s, ;  hence,  to  lose  Is,  takes  |  lb.  We  place  these  results 
in  Col.  2,  opposite  their  prices.  Reducing  the  fractions  in  Col.  1  and  2  to 
a  common  denominator  separately,  the  numerators  are  the  proportional 
parts  required.    Hence,  the 

Rule. — I.  Write  the  prices  of  the  articles  in  a  colinnn 
in  their  order,  with  the  mean  price  on  the  left. 

II.  Tahe  them  in  pairs,  one  less  and  the  other  greater 
than  the  mean  price,  find  how  inuch  is  required  of 
each  article  to  gaix  or  lose  a  unit  of  the  mean  price, 
and  set  the  results  in  Col.  1,  opposite  to  its  price.  Com- 
pare the  other  couplet  in  like  manner,  setting  the  results 
in  Col.  2. 

III.  Finally,  reduce  the  numbers  in  each  column  sep- 
arately to  a  common  denominator ;  the  numerators  will 
he  the  proportional  parts  required. 

Notes, — 1.  If  there  are  three  articles,  compare  the  price  of  the  one 
which  is  greater  or  less  than  the  mean  price  with  each  of  the  others,  and 
take  the  sum  of  the  two  numbers  opposite  this  price. 


380  Appendix. 

2.  The  reason  for  considering  the  articles  in  pairs,  oiie  above,  and  the 
other  below  the  mean  price,  is  that  the  loss  on  one  may  be  counterbalanced 
by  the  gain  on  another. 

3.  When  the  given  prices  are  integers,  the  same  results  are  readily 
found  by  taking  the  diflference  between  the  price  of  each  article  and  the 
mean  price,  and  placing  it  opposite  the  price  with  which  it  is  compared,  as 

4.  How  much  coffee  at  9,  11,  and  14  cents  a  pound,  will 
form  a  mixture  worth  12  cents  a  pound  ? 

5.  How  much  g-inger  at  15,  18,  21,  and  22  cents  a  pound, 
will  form  a  mixture  worth  19  cents  a  pound? 

912.  To  find  the  Other  Quantities  when  the  Mean  Price  of 
the  Mixture  and  the  Quantity  of  one  of  the  Articles  are  given. 

6.  How  many  pounds  of  starch  worth  11  and  15  cents  a 
pound,  must  be  mixed  with  16  lb.  at  10  cents,  so  that  the  mix- 
ture may  be  worth  13  cents  a  pound. 

Analysis. — If  neither  article  were  limited,  opebation. 

the  proportional  parts  would  be  2,  2,  and  5. 
(Art.  911,  N.  1.)  But  the  quantity  at  10  cts.  is 
limited  to  16  lb.,  and  its  proportional  part  is 
2  lb.  Now,  16-i-2  —  8.  Therefore,  multiplying 
each  of  the  proportional  parts  by  8  gives  16 ; 
16  and  40  lb.  the  mixture  required.     Hence,  the 

KuLE. — Fijid  the  proportional  parts  as  if  the  quantity 
of  neither  article  were  limited.     (Art.  911.) 

Divide  the  limited  quantity  hy  its  proportional  part, 
and'  multiply  each  part  found  hy  this  quotient;  the 
product  luill  he  the  quantity  required. 

Note. — When  the  quantities  of  two  or  more  articles  are  given,  find  the 
amrage  value  of  them,  and  considering  their  sum  as  one  quantity,  proceed 
as  above. 

7.  How  much  corn  at  45,  56,  and  65  cents  per  bushel,  must 
be  mixed  with  25  bu.  of  oats  at  40  cents,  so  that  the  mixture 
may  be  worth  50  cants  a  bushel  ? 


Col.  1. 

2. 

10 

2 

2 

13 

11 

2 

2 

15 

3+2 

5 

N  S  W  E  R  S . 


Article  50. 

19.  1418. 

Arts.  70,  71. 

31.  2280. 

12.  8879. 

13.  7889. 

14.  7979. 

15.  9798. 

16.  .S6178. 

17.  966  mi. 

18.  987  Acres. 

20.  1836. 

21.  ir83. 

22.  2604. 

23.  7512. 

24.  21241. 

25.  10562. 

26.  2742. 

2.  10292. 

3.  10083. 

4.  27886. 

5.  34339. 

8.  216.9. 

9.  182.19. 
10.  $242.19. 

32.  5583. 

33.  7271. 

34.  84841. 

35.  5482. 

36.  14935. 

37.  985. 

38.  7065. 

27.  2355. 

28.  $627. 

11.  $3684.939. 

39.  $301.12. 

40.  $5072.35. 

Arts.  55,  5f), 

29.  630  lb. 

2.  12054  yd. 

3.  14792  rods. 

30.  ij^3789. 

31.  $3582. 

32.  $1323. 

33.  $279075. 

34.  .■>595522. 

Art.  72. 

1.  113  yd 

Art.  74. 

1.  439.25. 

4.  19747  wk. 

2.  $221. 

2.  $1291. 

5.  28143  lb. 

3.  189  gents. 

3.  83412.7. 

7.  415.034. 

35.  $2115306. 

4.  1003  bu. 

4.  $72320. 

8.  114.634. 

36.  $1606895. 

5.  374  bu. 

5.  $985.25. 

10.  $8S.967. 

37.  7448208453. 

6.  $1989. 

6.  146  trees. 

11.  $104,721. 

38.  $.^068,  farm. 

7.  $479. 

7.  $1090. 

12.  $100.84. 

$6136,  all. 

8.  $1659. 

8.  12520  bu. 

13.  $93,833. 

39.  .^16646. 

9.  $3023. 

9.  $1910.89. 

40.  $37650. 

10.  $1763. 

10.  $5491. 

Art.  60. 

41.  1727. 

11.  $3747. 

11.  $5250. 

42.  $8475. 

12.  1825. 

12.  627067. 

1.  $6821. 

43.  14,507,407. 

13.  2600. 

13.  $21422. 

2.  $2324. 

44.  7,597,197. 

14.  3085. 

14.  22225. 

3.  $4900. 

45.  17,364,111. 

15.  1306. 

15.  16,014,400. 

4.  $1444. 

46.  8,919,371. 

16.  4098. 

16.  184.815,000, 

5.  503  trees. 

47.  1,767,697. 

17.  1108. 

000. 

6.  73  yr. 

48.  50,155,783. 

18.  4531. 

17.  1486.75. 

7.  $1648. 

19.  14520. 

18.  5  times. 

8.  $34950. 

20.  24622. 

19.  61483.95. 

9.  $33700. 

Art.  69. 

21.  125028. 

20.  26973. 

10.  565. 

22.  64303. 

21.  34059.5. 

11.  742. 

2.  3232. 

23.  224066. 

22.  1912,  B's. 

12.  1530. 

3.  3244. 

24.  103S75. 

4482,  C's. 

13.  1779. 

4.  3424. 

25.  420486. 

23.  5986. 

14.  1597. 

5.  3525. 

26.  $16014. 

24.  33086330. 

15.  1757. 

6.  3213. 

27.  $1315. 

16.  2379. 

7.  501  Acres. 

28.  $5385. 

Arts.  S4,  85. 

17.  2619. 

8.  $1134. 

29.  708. 

18.  1020. 

9.  412292. 

30.  942. 

2.   17501b. 

382 


Answers. 


10. 

11. 

12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 
22. 

24. 
26. 


3.  4410  sheep. 

4.  2022  1b. 

5.  3000  ft. 

6.  4345  yd. 

7.  12768  bu. 

8.  20712  in. 
1924.5.- 
402.12. 
434.98. 
60.221. 
787.14. 
$26116.02. 
$3381.19. 
-150981.28. 
$74241.84. 
$7264,854. 
$138.24. 
$1455.78. 
$7.68. 
$7,857. 
$60. 
.^3000. 


27.  $42120. 

28.  $6000. 

29.  $45.36. 

30.  $21150. 


1. 
2. 
3. 
4. 

5. 
6. 


9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 


Art.  87. 

1445. 
$23646. 
$173.34. 
996.84  lb. 
234588  yd. 

Art.  88. 

16425d. 
91350  lb. 

$8991. 

$16884. 

68520  ft. 

1564  sheep. 

$6000. 

$17920. 

17945  bu. 

48000  A. 

20(i952. 

98982. 

204336. 

368109. 

$1849.65. 

$3201.44. 

438480. 

$1443.099. 

31,968,868. 


30.  48,053,208. 

21.  34,628,175. 

22.  65,404,110. 

23.  639,756. 

24.  1560.975. 

25.  3,071,926. 

26.  3,007,368. 

27.  24,631,008. 

28.  35,497,655. 

29.  42,546,240. 

30.  62,355,319. 

31.  849,126,321. 

32.  1.219,641,537. 

33.  1,988,907,892. 

34.  2,758,104,145. 

35.  $576. 

36.  $3744. 

37.  6930  mo. 

38.  58050  ds. 

39.  $632. 

40.  $149. 

41.  1549.25. 

42.  $488.25. 

43.  $42255. 

44.  $1895. 

45.  168750  1b. 

46.  $57649,  cost. 
$3367,  dif. 

47.  $534842. 

48.  $618. 

49.  $22. 

50.  $56.50. 

51.  1624  trees. 

52.  1584  pupils. 

53.  62234. 

54.  20,345,400. 

55.  1776  bu. 

56.  368,640. 

57.  283,410. 

58.  270,592. 

59.  111,168. 

60.  92,538. 

61.  493,480. 

Art.  89. 

2.  $2295. 


3. 

4.  $684. 

5.  8610  miles. 

6.  $4950. 

7.  $312. 

8.  25760  bu. 

9.  $16128. 
10.  $91080. 


Art.  91. 

6.  468001b. 

7.  809600  pp. 

8.  476,000. 

9.  534,860,000. 

10.  1,204,670,800,000. 

11.  26,900,785,000,000. 

12.  890,634,570,000,000. 

13.  946,030,506,800,000. 

14.  3,840,000. 

15.  10,940,000. 

16.  2,075,994,000. 

17.  390,677,500,000. 

18.  372,000. 

19.  11,840,000. 

20.  373,520,000. 

21.  3,603,200,000. 

22.  55,447,000,000. 

23.  37,800,000,000. 

24.  25,800,000,000. 

25.  4,059,360,000. 

26.  14,760,000,000. 

27.  6,204,000,000. 

28.  1,672,650,000,000. 

29.  1,075,635,900,000. 

30.  450,230,874,000. 

31.  6,980,161,370,000. 

32.  834,271,780,000. 

33.  779,984,000,000. 

Art.  93, 

17.  103  cts. 

18.  3600  soldiers. 

19.  $18200. 

20.  452  miles. 

21.  $504. 

22.  $1492. 

23.  $4632. 

24.  $234. 

25.  8288d.  s.  q. 

26.  1460  mi. 

27.  2,015,028. 

28.  8,498,120. 

29.  404,444,040. 

30.  6,342,737,821. 

31.  351,039,462,230. 

Arts.  107,  110. 

3.  1275. 

4.  1173. 

5.  2468. 

6.  1317. 

7.  11449. 


Answers, 


383 


8.  11155. 

10.  378. 

11.  246. 

12.  427. 

13.  1234. 

14.  546. 

15.  1234. 

16.  335. 

17.  349. 

19.  913|. 

20.  661|. 

21.  820|. 

22.  4639 1. 

23.  15290|. 

24.  20588|. 

25.  8731i. 

26.  9124f. 

27.  120421f. 

28.  71410|. 

29.  96043i. 

30.  87105|. 

31.  82401 . 

32.  8369f. 

33.  56234. 

34.  1533. 

37.  640.87. 

38.  80.666. 

39.  24631  yd. 

40.  $5.42. 


Art,  112, 

2.  2862f|. 

3.  12431^^. 

4.  13967Jy. 

5.  29493tV 

6.  IO9O3V 

7.  1231|f. 

8.  192.3011- 

9.  24218|f. 

10.  $1.20||. 

11.  $1.27^. 

12.  $1.3541^. 

13.  $2.007i|. 

Art.  115, 

1.  61  shares. 

2.  31||yr. 

3.  48|ilihd. 


4. 
5. 
6. 

7. 


21. 
22. 
23. 
24. 

25. 
26. 

27. 
28. 
29. 


43ff  mi. 
75  dresses. 
51||  m. 
$50. 
73.6  A. 
45  If  casks. 
27  days. 
$31.14i§. 
$3.15i|. 
37.4^V 


9. 
10. 
11. 
12. 
13. 
14. 

15.  199.2 
16. 
17. 
18. 

19.  $1.36|i. 
20. 


1.463V 


340ff. 

175ff. 

65^^ 


$1.53i|. 

84/^. 

149f|. 

837ff. 

1607f|. 

962||. 

1006ff. 


901b. 

14911  A. 

126  boxes. 
30.  288  eggs. 
32.  245iif. 
33. 
34. 
35. 
36. 
37. 
38. 
39. 


133HI. 

9633«o^e- 
720|ff. 
3011fa|. 

3938IM. 
6671fti. 


7318ff|. 

40.  121.93f|-|. 

41.  3.1885fi|- 

42.  673.888|ff. 

43.  456.607iW7- 

44.  2680.52.^111. 

45.  2.3631tt|f|. 

46.  5.2558f4|||. 

47.  5.109|fif|. 

Arts,  117,  118, 

2.  3452  and  31  rem. 


3.  672  and  487  rem. 

4.  642  and  3544  rem. 

5.  73  and  64159  rem. 

8.  340  bar. 

9.  $456.50. 

10.  80  bales. 

11.  $40. 

12.  292^^  lb. 

13.  4160  men. 

Art.  121. 

1.  14. 

2  12 

4!  100  s.  150  g. 

5.  118  B.,  155  A. 

6.  69  cts. 

7.  82  vr. 

8.  $283. 

9.  392  mi. 

10.  1  mi. 

11.  $5243,  B. 
$17176,  C. 
$23684,  all. 

12.  $621. 

13.  730  sch. 

14.  $20  per  A. 
$1868,  gain. 

15.  $66.52|i|. 

16.  Cows;  $6094. 

17.  228  A.,  B.'s. 
114  A.,  C.'s. 
1710  A.,  all. 

18.  18,  sm.  No. 
882,  gr.  No. 

19.  406  oxen. 

20.  38,818,897,  dif, 

21.  $3780. 

22.  83. 

23.  105374. 

24.  14. 

25.  1213. 

26.  43. 

27.  8117. 

28.  723.8. 

29.  49312  mem. 

30.  3082  men. 

31.  $718. 

32.  196  sofas. 

Art,  125o 

6.  112. 

7.  120. 

8.  13f. 


384 


Answers. 


9. 

5h 

10. 

13. 

11. 

22i. 

12. 

mh 

18. 

3. 

14. 

Wt- 

15. 

15  tons. 

17. 

Ifi  bags. 

18. 

33f  bar. 

19. 

126  bar. 

Art.  143. 

2. 

5x5x3x3. 

3. 

47  X  2  X  2  X  2. 

4. 

48  X  2  X  2  X  2. 

5. 

3x3x2x2x2x2 

x2x2. 

6. 

7x3x2x2x2x2 

x2. 

7. 

199  X  2  X  2. 

8. 

3x8x3x2x2x2 

x2x2. 

9. 

7x5x3x8x3. 

10. 

19  x5x  3x3x2x2 

11. 

37x5x5x5x2x2 

12. 

67  X  43  X  2  X  2  X  2 

x2. 

18. 

6029  x2x2x2x2. 

14. 

1297  X  2  X  2  X  2. 

15. 

5x5x2x2x2x2 

x2x 2x2x2x2 

x2. 

16. 

508x2x2x2x2 

x2x2x2 

17. 

193  X  2  X  2  X  2  X  2 

x2x2x2x2x2 

ArU  14:4, 

19. 

2  and  2. 

20. 

2. 

21. 

2. 

22. 

2. 

23. 

2,  8,  and  7. 

24. 

2,  2,  2,  and  3. 

25 

2,  2,  and  2. 

26 

5  and  5. 

27 

2  and  2. 

Art.  149. 

2 

21. 

3 

13. 

4 

.  19. 

5.  15. 

7      22B 
'  •     2  8  8* 

6.  3. 

8..  Iff. 

7.  4. 

8.  12. 

9.  iff. 

9.  0. 

10.  5. 

Art.  185^ 

Art. 

150. 

2.  fi. 

3    ifi 

1.  6. 

A       16 
*•     25- 

2.  15. 

5.  \. 

3.  12. 

6.  tV 

4.  1. 

7.  iM. 

5.  5. 

6.  4. 

8.  ii. 

7.  4. 

Q      22 

8.  4. 

10.  f. 

9.  8. 

10.  4bu. 

11.  4  A. 

11    i|t 
12.  f^. 

12.  21. 

13.  J^. 

13.  60ft.  wide;    10  1., 

14.  |. 

2  1.,  and  15  1. 

15.     TT- 

14.   $252,  price  ;   5  li.. 

1  1 

16    1 

9  h.,  and  11  b. 

J-U.       5. 

17       431 

Art. 

157. 

2.  240. 

^?^     i«^ 

8.  12600. 

1.  37i 

4.  504. 

2    44-*- 

5.  1134. 

/O.     '±'±13. 

6.  144. 

3.  30. 

7.  130645. 

4  28i. 

8.  533610. 

5.  31i. 

9.  156240. 

6.  53V 

10.  144. 

7.  14l|. 

11.  2520. 

8.  28#8. 

12.  262080. 

13.  1921506000. 

9.  22|ft. 

14.  360. 

10.  46tV 

15.  1584  ft. 

11.  21t%V 

16.  $60. 

12    lOA^TT. 

17.  840  gal 

13.  40i|f. 

18.  12  hr. 

19.  24  hr. 

14.  993^^. 

20.   120  br. 

15.  13^V 

16.  110^. 

Art. 

182. 

17.  131iA|. 

9        40 

18.  l|i 

q      65 

19.  SIMM- 

4.  m- 

20.  12TV¥r- 

5.  m- 

21.  60if  rd. 

6.  m. 

22.  $153J3VV     c 

\ 


■x/ 


Answers. 


385 


Art,  ISO, 

9. 

0  67 

2. 

3. 

69 

10. 

9^. 

214 

11. 

41. 

4. 

968 

12. 

A5  8  9 

^sjo- 

5. 

18. 

1119 
l^T^- 

6. 

3 

231 
5^  * 
4980 

313 

•\¥- 

2543 

14. 

41tV. 

7. 

15. 

^^h- 

8. 

16. 

^\U- 

9. 

17. 

m.]io- 

10. 

18. 

111  62  9 

11. 

19. 

1389 

12. 

5   * 

57  6 

ill45_ 

20. 

651^0. 

13. 

21. 

199if  i  lb. 

15 

22. 

289|ii  ra. 

Art.  195, 

23. 

553/,-  yd. 

2. 

3. 
4. 
5. 

88   30 

25. 

Uf. 

3  5   2  7 

26. 

16|. 

¥5»  T5- 
3  0   5  6 

27. 

33i§. 

^0'  ^0- 
4  8    6  3 

28. 

16H. 

ITT'  TT4- 

29 

mi 

6. 

15  7500    113400 

2S3oO(J»    2  8  3  5  0  0' 
19845 

30. 

37H. 

■J8¥50(T- 

31. 

222 

7. 

120528     208656 

--"^s* 

3^657^'    3^6o92^' 
96768 

32. 

oqi  1 

3¥65¥^- 

33. 

5I5V 

34. 

16511  yd. 
297f  J  m. 

Art,  196, 

35. 

9. 

2  4   2  5   16 
60'  ^0'  60- 

10. 

264   385   234 
^T6'  ^T^'  1?T^- 

Art,  204 

11. 

45    32    15 

T(J^'  TTJ^'  TTTS"- 

8. 

4079 

ssso- 

12. 

735   600   672   490 

¥TT>  ?'T0'  ^TTJ'  8TT5' 

4. 

1  6 

T^I-- 

13. 

189   180   168    8  4 

2TI7'  ^T0»  2T(y>  "JTo- 

5. 

4  9  1 
T650- 
02 

5f. 

14. 

672    1240    94  5 
T40  0'  ITOI^'  TIOO"- 

6. 

15. 

210   395   1092 
5^5»  o^Z'      F55  • 

7. 

16. 

3  5    9  15    12 

TT)F'  TF5>  TOI^- 

8. 

3^^- 

17. 

12    175   680 
■^80'  2B^0'  2?(J- 

9. 

8 
77* 

18. 

154   2590   180 

2T5'   245"'  2T5- 

10. 

1 
•>  • 

19. 

972   189   100 
132'  T32'  132* 

11. 

3H- 

20. 

117    85    63 
11^9'  T89'  189- 

12. 

13. 

If- 

Art,  201, 

14. 

391 

3. 

1JL9_ 

J^  1  6  5  • 

15. 

llfi- 

4. 

0653 

~  7  2  8  • 

16. 

n- 

5. 

Q2  4  3 

^2  8  0- 

17. 

mh 

6. 

1t¥77. 

18. 

317|i  rd. 

7. 

51. 

19. 

357:j?T7  T. 

8. 

21. 

20. 

303 1  lb. 

21. 

131611  bu. 

22. 

|15|. 

23. 

mil  A. 

24. 

38U  yd. 

Art,  200, 

8. 

If. 

4. 

i^V 

5. 

4. 

6. 

5|. 

7. 

7. 

8. 

28|. 

9. 

4^r. 

10. 

5. 

11. 

5. 

13. 

702. 

14. 

1988. 

15. 

4941. 

16. 

3537. 

17. 

769^. 

18. 

27612. 

Alt,  208. 

2. 

6|. 

3. 

A 

4. 

45. 

5. 

174. 

6. 

60." 

7. 

255. 

8. 

36. 

9. 

432. 

10. 

1191. 

12. 

1178. 

13. 

8450. 

14. 

1280. 

15. 

341111 

16. 

4496. 

17. 

8118. 

18. 

1042811. 

19. 

5611|i 

20. 

5086^%. 

21. 

43452. 

22. 

74269 i|. 

23. 

91806. 

Art,  211, 

5. 

1 

3- 

6. 

3 
7- 

7. 

21 

8. 


386 


Answers. 


5(?- 
_3_ 
35- 
rc  5 


9. 

10. 

11. 

12.  11  If. 

13.  2UV 

14.  7ii. 

15.  $1031. 


1.  $12||. 

2.  136  cts. 

3.  112i  cts. 

4.  23r)~cts. 

5.  mi 

6.  33  L  cts. 

7.  292.1  cts. 

8.  344]  cts. 

9.  $57if. 

10.  630  cts. 

11.  1161 

12.  $4|.' 

14.  $5i^i. 

15.  $5if. 

16.  12371  cts. 

17.  4061  "cts. 

18.  300;^  cts. 

19.  806i  cts. 

20.  $7. 

21.  $10if. 

22.  273f. 

23.  .f3«\. 

24.  $41|t. 

25.  621-1%  cts. 

26.  $831^3. 

27.  391f  cts. 
2S.  $1331 
29.  $65}f. 
80.  $615/^. 

31.  743|  m. 

32.  2310. 

33.  32|. 

34.  197f. 

#  .>  6  t> 

TUIT975T' 

147 

¥67- 


38. 
39. 
40. 
41. 
42. 
43. 
44. 


867 

o 

7- 

80. 

1561^11 

109A«s. 

J>-22_ 
1325* 


6-" 


60- 


Art,  215. 

2.  H. 

t5.  3  1. 

4  _7JL 

^-  7  5  6' 

O.  7j. 

6  109 
"•  3T50' 

7  __L7_1 
'•  1850* 

Q  29 

'^-  7  7  5- 

10  -8  '  9 

^^-  2394* 

^**  5625* 

1R  4 

15.  l\  T.;  1  part. 

16.  |if. 

2.  294. 

3.  432. 

4.  576. 

5.  1350. 

6.  48. 

7.  171. 

8.  1344. 

9.  804. 

10.  11045. 

11.  461yd. 

12.  144  d. 

14.  9i 

15.  18. 

16.  411f. 

17.  81614. 

18.  12^.^ 

19.  22f. 

20.  480  sh. 

21.  56{|yd. 

22.  34 1  cloaks. 

23.  52  c.  3  rem. 

24.  30x\  br. 


Art,  220, 

4   4^ 
5.  20. 

6. 

7. 


1  _? J5  3_ 
■^1096- 
4 


Art,  221. 


1. 

5i^lb. 

2. 

10^  c. 

3. 

8y\v  bar 

4. 

%^jh- 

5. 

7  cts. 

6. 

Q  44     o 

7. 

p^m- 

8.  16. 

q      (feO  fi  3  3 

10.  llf^  T. 

11.  87t^V  sacks. 

12.  157y%V  bales. 

13.  i^%,.  " 

15      _38_ 
^^'     1T05' 

16.  f. 

1 7  -8J 

18  7  '•-!- 
19.  171. 
30.  tI  r. 

22  5^^ 

"•*•  •-'3  4' 
23.     y|^. 

24.  ff|. 

25.  M. 

26.  3V7. 

27  1_7  7 

28  19 "9 

'^*-'-     1  9  8  4  • 
1058 
¥"(>T5- 


3.  If. 

4.  tV 

5.  1^. 

6.  2^0- 

7    1  J. 

Q      41 

9.  U. 


10.  1^. 


11. 

163 

20G* 

12. 

313|. 

13. 

805 
8  3  7' 

14. 

33^^' 

15. 

2  58 

16. 

1V0-. 

17. 

167 

iso- 

18. 

ItV 

19. 

451 
6T8' 

20. 

'iOfil4  3 

Art.  226, 


3. 
4. 
5. 
6. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 


1 

^^• 

_7 
496* 

11 
T¥3^- 

19 
TT5' 

1 
T40' 

7 

2600* 
J)_0  6 
1  5  7  9" 

5 
3F- 

_2JL 
477- 
2 
3- 
10 
IT- 
113 
280* 
1  7     A 
^3    ^• 
30 
3  7' 
2^7 
38* 
65 

m- 

3 

8* 

3  515 

53  7  3- 

__3_ 

20- 


Art,  228, 

2.  64. 

3.  70. 

4.  135. 

5.  160. 

6.  165. 

7.  2891. 

8.  844f. 

9.  1249  vV 

10.  $14560. 

11.  $7080. 

12.  315  s. 

13.  $455. 


Aiiswers. 


387 


Art,  230. 

-(  117 

^-  ¥¥3- 

2  729 

•  T2TT- 

3.  2x3x3x3x3x8. 
2x2x2x3x3x2 

x3x3. 
3x2x3x3x3x3 
x8. 

4.  200,  1.  c.  d. 

5.  6^^ 
G.  4i|. 

7.  1011  J. 

8.  iileft. 

'f-  so- 
lo. 411|m.,  both  go. 
401  ra.,  (jif. 

11.  $48}. 

12.  lU. 

13.  13|. 

14.  17tV. 

15.  32. 

16.  22|. 

17.  39tV 

18.  134^5^. 

19.  249^^. 

20.  $20|. 

21.  $130|f. 
23.  $29056}. 
23.  112||. 

O/l  3  5 

'*"*•  333- 

25.  53V 

26.  $12521^. 

27.  $18521^1. 

28.  S3212}f§. 

29.  $2681i. 

30.  15|  lots. 

31.  $32yVlost. 

32.  Hi  books. 

33.  $82|. 

34.  $75^. 

35.  305  eggs. 

36.  $12250. 

37.  $36752. 

38.  $1000. 

39.  64  ft. 

40.  75  ft. 

41.  102  marbles. 

42.  $112i 


43.  108  ds.  C. 

44.  15360  men. 

45.  100. 

46.  100. 

47.  38. 

48.  13|. 

^O.      2  9- 

50.  3H. 

51.  $3311. 
53.  131  oz. 

53.  $3808. 

54.  $4750. 

Art,  245, 

3.  .3000,3.0004,7.0080. 

3.  .300,  .060,  .008. 

4.  .0300,  .1350,  .7000, 

.3363. 

5.  .3600,  .3750,  .0336, 

.3060. 

6.  .0450,  .6100,  .0035, 

.1080. 


Art,  24:7. 


3  -'- 

4.  14 
^*  2  5' 

5.  I 

6.  f. 

7.  Jo. 

Q  5 

10  -5- 

nl 

12. 

13.  1 


_3 

5  0  07' 


3  1_ 
125- 

4  100  0  7_ 
TOOOITOO' 


14 
15 
16 

17      -33 
^*-     4  OTIT- 
IS     i^ 


5000000' 


Art,  249, 


3.  .35. 

3.  .75. 

4.  .635. 

5.  .8. 

6.  .135. 

7.  .375. 

8.  .75. 


9.  .875. 

10.  .45. 

11.  .35. 
13.  .4. 

13.  .65. 

14.  .43. 

15.  .033. 

16.  .09375. 

17.  .33UV. 

18.  75.6." 

19.  89.75. 

30.  39.635. 

31.  65.135. 
33.  8.075. 

33.  .83. 

34.  27.8125. 

25.  93.1875. 

26.  .66666|. 

27.  .33333.1. 

28.  .44444|. 

29.  .16379:t%. 

30.  .13666|. 

Art,  254, 

2.  $182,731. 

3.  238.4313. 

4.  1073.16387. 

5.  386.5134. 

6.  133.3387. 

7.  $411,135. 

8.  $738. 

Art,  25(i, 

3.  $3,635. 

4.  $1743.35. 

5.  $5.75. 

6.  $1305.90. 

7.  $170,675. 

8.  $533.65. 

9.  •806.41m. 

116.09  m. 

Arts.  259,  2 GO. 

3.  .135918. 

4.  16.141753. 

5.  37.478884. 

6.  441.3336144. 

7.  813.475. 

8.  5151. 

9.  164.0635. 

10.  1503.4375. 

11.  47.0331027. 


388 

12.  .1719375. 

13.  3G75. 

14.  17.91)53125. 

15.  .00187441562. 

16.  2374.2125. 

17.  14G47.5  cts. 

18.  53200  cts. 

19.  $3803.625. 
21.  46384.2. 
2.>.  6423.02. 

23.  25460. 

24.  3004. 

Art.  263, 

4.  .9128  +  . 

5.  .9408  +  . 

6.  .008911  +  . 

7.  .09. 

8.  10. 

9.  .000102  +  . 

10.  3.4542  +  . 

11.  846.37105 +  . 

12.  .116|. 

13.  .7. 

14.  9.991.2844 +  . 

15.  8930. 5972  +  . 

16.  761409.375. 

17.  73.79512  +  . 

18.  124.33^. 

19.  48  coats. 

20.  8  yd. 

21.  15.15+  hr. 

22.  56  bar. 

23.  4.637+  d. 

Art.  264, 

25.  .2425. 

26.  .45631. 

27.  .032463. 

28.  .00008534. 

29.  .00642564. 

30.  .5634527. 

Art.  279, 

1.  $30.68. 

2.  $12.13. 

3.  $45,805. 

4.  $196.51. 

5.  $362.50,  cost. 
$71.50,  dif. 

0.  $16,865,  dif. 

7.  $93,625. 


Anstvers. 

8. 

$471.25. 

9. 

$108.75,  gain. 

10. 

$1080. 

11. 

$1601. 

12. 

$1180. 

13. 

$8429,  gain. 

14. 

$7540. 

15. 

5  liats. 

16. 

35. 65 If  w. 

17. 

$2,574  +  . 

18. 

465.551  bii. 

19. 

$10,282. 

20. 

$136,986  +  . 

21. 

9  yd. 

22. 

252  bar. 

23. 

$.125  +  .  . 

24. 

600  T. 

Art,  284, 

2. 

$438. 

$300. 

4. 

$137.60. 

5. 

$112.25. 

6. 

$250. 

8. 

$1061.33i 

9. 

$364. 

10. 

$300. 

Art,  285, 

12. 

1700  yd. 

13. 

450  lb. 

14. 

1950  bu. 

15. 

962  cans. 

16. 

128  yd. 

17. 

132  hoes. 

Art,  286, 

20. 

$355.25. 

21. 

$()91.08. 

22. 

$315.75 

23. 

$4733  82. 

24. 

$111.65. 

25. 

.1S44.10. 

26. 

$100.49. 

27. 

$1866.90. 

28. 

$191.93. 

Art.  287, 

31. 

$68.31. 

32. 

$19,642. 

33. 

$652  03125. 

Art,  299, 

1. 

$17.77. 

o 

$158.256|. 

3! 

$360,705. 

4. 

$416.02. 

5. 

$2659.275. 

6. 

$1067.65. 

7. 

$429.8825. 

8. 

$372,755. 

9. 

$1058.05. 

Art,  331. 

2. 

105.735  m. 

4. 

64  liters. 

6. 

3.990  Km. 

7. 

$240.50. 

8. 

$54. 

10. 

26|  Km. 

11. 

174.52592  Km 

12. 

$4.50. 

13. 

54  cts. 

14. 

$819. 

15. 

$1.15. 

16. 

$23,125. 

17. 

$105.60. 

18. 

$2.40. 

19. 

21  kilos. 

Art.  332, 

21. 

3  sq.  m. 

22. 

37.8  sq.  m. 

23. 

85.2  sq.  m. 

24. 

1044  centars. 

Art.  333. 

26. 

853.632  cu.  m. 

27. 

768  cu.  m. 

28. 

$87. 

29. 

$12,375. 

30. 

$59.8598. 

Art.  396, 

0 

26612  ft. 

3. 

825264  gr. 

4. 

434035  lb. 

5. 

3393088  oz. 

6. 

121528.5  ft. 

7. 

140304  gr. 

8. 

1539776  oz. 

9. 

10524  gi. 

10. 

42960  pt. 

11.  2936  pt. 

12.  161856  sq.  in. 

13.  1395553.5  sq.  ft. 

14.  3335  cu.  ft. 

15.  364620  min. 

16.  184584  Ixr. 

17.  522028  sec. 

18.  16362  d. 

19.  197150  far. 

20.  120475020  sec. 

21.  119.84. 

22.  $10.92,  gain. 

23.  $2597.82. 

Art.  400, 

3.  33  bbl.  30  g.  3  qt. 

4.  92hlid.ll»bl.27gal. 

2qt. 

5.  27231b.7oz.  I4pu^. 

6.  74  lb.  11  pwt.  4  gr. 

7.  22  cwt.  26  lb.  8  oz. 

8.  263  T.  2  cwt.  95  lb. 

4  oz. 

9.  150  rd.  2  ft.  4  in. 

10.  9  m.  880  ft. 

11.  313sq.rd.49.75sq.ft. 

12.  437  A.  102  sq.  rJ. 

13.  129  C.  56  cu.  ft. 

14.  1350  bu.  28  qt. 

15.  452  bu.  14  qt. 

16.  649  com.  vr.  20  da. 

17.  122385  wk.  5  d. 

18.  £4878,  7s.  8d. 

19.  $6.90. 

20.  $750.75. 

21.  $1.89. 

Art,  402. 

3.  9  hr.  20  min. 

4.  5  d.  14  lir.  24  min, 

5.  3  fur.  22  rd.  3  ft. 

Sin. 

6.  1  pk.  5  qt.  12  pt. 

7.  274  A.  45  sq.  rd. 

21-i|  sq.  yd. 

8.  I  of  a  gill. 

9-  iH,  pt. 

10.  9s.  3d. 

11.  3  qt.  .048  pt. 

12.  15  hr.  34.56  sec. 

13.  85  lb.  9.6  oz. 

14.  3  pk.  .5248  pt. 

15.  5  yd.  1  ft.  2.04  in. 


Ansivei's. 

Art.  403, 

19. 

If  bu. 

20. 

f  i  gal. 

21. 

^¥o%  wk. 

22. 

^oV(j  lb. 

23. 

47           A 

24. 

.828125  bu. 

25. 

.75625  d. 

26. 

.88^  yd. 

27. 

.697911  lb. 

38. 

.05  gal. 

29. 

.000125  T. 

30. 

2.3  yr. 

Art,  404, 

31. 

12 
^3* 

qo 

14 

O/J. 

12T- 

83. 

195 

¥8".r- 

34. 

.12648  +  . 

35. 

.405  +  . 

Art.  405, 

3. 

39.1482  mi. 

4. 

19.8131  gal. 

5. 

15.89  bu. 

6. 

4.2324  oz. 

7. 

303.68365  lb. 

9. 

148.87775  A. 

10. 

4287.92  cu.  ft. 

Art,  40(L 

12. 

58.293+  m. 

18. 

6286.959+  Kg. 

14. 

236.58559+  li. 

15. 

72.497  +  kl. 

16. 

143.228+  Kg. 

17. 

6000  sq.  m. 

18. 

16.378+  hektars. 

19. 

410.748+  cu.  m. 

20. 

27985.715+  cu.  m 

Art,  407, 

2. 

£29,  7s.  Id. 

3.  23  gal.  2  qt.  1  gi. 

4.  16  wk.  6  da.  4  hr. 

48  min. 

6.  184  bu.  3  I'k.  7  qt. 

7.  249  A.  157  sq.  rd. 


389 

8.  6  hhd.  53  gal.  3  qt. 

9.  200yr.  llmo.Owk. 

4  da. 

10.  101  mi.  160  rd. 

11.  109  sq.  yd.  8  ft. 

142  in. 

12.  73  C.  69  ft.  177  in. 

13.  177mi.  242rd.  4yd. 

2  ft.  4  in. 

15.  lbu.2pk.  lqt.|Jpt. 

16.  9  hr.  37  min.  25?  J 

17.  8  oz.  3  pwt.  22.4  gr. 

18.  5  d.  16  hr.  6  min. 

51  f  sec. 

19.  1  gal. 

20.  18s.  5d.  2!  far. 


Art,  40 S, 


2. 

58  lihd.  6  gal.  2  (p. 

3. 

6  oz.  18  pwt.  2  gr. 

4. 

113^  yd. 

5. 

9  mi.  0  fur.  18  rd. 

7  ft.  10  in. 

6. 

54  A.  149  rd.  38  ft. 

7. 

128cu.ft.l652cu.in. 

8. 

48  C.  106  ft.  58  in. 

10. 

3  yr.  5  mo.  21  d. 

11. 

6  yr.  4  mo.  26  d. 

12. 

12  yr.  2  mo.  28  d. 

13. 

1  yr.  7  mo.  21  d. 

14. 

5  yr.  4  mo.  15  d. 

16. 

191  d. 

17. 

150  d. 

18. 

74  d. 

19. 

150  d. 

20. 

222  d. 

21. 

10=  54'  13'^ 

22. 

98'^  0'  57". 

23. 

19^  r  54". 

24. 

79^  20'  15". 

Art.  410, 

2. 

68  lb.  1  oz.  4  pwt. 

3. 

195  gal.   8  qt.  0  pt. 

Igi- 

4. 

286mi.  150rd.  3  yd. 

1  ft.  6  in. 

5. 

8  1b.  10  oz.  7  pwt. 

6. 

1538  gal.  1  qt. 

7. 

532mi.  Ofur.  lOrd. 

390 


Answei's, 


Art,  411, 

2.  5  oz.  8  pwt.  8  ^Y. 

3.  10  lb.  lU  oz. 

4.  4  m.  177  r.  7  ft.  6  in. 

5.  19  bu.  0  pk.  2  qt. 

Alt.  415. 

2.  2°  15'  30''. 

3.  10°  53'  1.95". 

4.  11°  59'. 

5.  9°  17'  6". 

6.  12°  20'. 

Art,  416. 

2.  29  mill.  35.2 1  sec. 

3.  41  min.  18/-  sec. 

4.  5  h.  45  m.  16.6  s. 

5.  1  lir.  6  min.  30  sec. 

6.  40  min.  28  sec. 

7.  12  min.  13  sec. 

8.  Ihr.l7min.  20isec. 
3  hr.  21  min.  |  sec. 

9.  N.  Y.,  9  A.M. 
Rich.,  Va.,  8  h.  46 

min.  29  sec,  A.M. 
San  Fr.,  5  h.  46  min. 
25  sec,  A.M. 

Art,  424. 

3.  36|yd. 
8.  48f  yd. 

4.  18  A.  44  sq.  rd. 

5.  435.6  ft. 

6.  $10890. 

7.  ^1^1980. 

8.  20358  sq.  meters. 

9.  3.25  meters. 

10.  $2. 

11.  640  A. 

12.  26  A.  65  sq.  rd. 

13.  50  rd. 

14.  11.6973  Ha. 

15.  5  A.  159  r.  260}  sq.ft. 

16.  2  sq.  rods. 

17.  20  A.  1120  sq.  ft. 

18.  $57500  gain. 

19.  $76. 

20.  $47,782. 

21.  $9498  50. 

22.  1330.56  tiles. 

23.  $11760. 


24.  120  rd.  wide. 

$2560.  cost. 

25.  41^  planks. 

Art.  429, 

2.  4200  cu.  ft. 

3.  23328  cu.  ft. 

4.  377 1  loads. 

5.  $125.92 If. 

6.  $2823.331. 

7.  198  cu.  yd.  13  cu.  ft. 

648  cu.  in. 

8.  137  cu.  yd.  24  cu.  ft. 

1512  cu.  in. 

9.  30  ft. 

Art,  431. 

1.  6f*6  cords. 

2.  14^6  cords. 

3.  $13.8125. 

4.  48  cu.  ft.  ;  80  cu.  ft. 

5.  1536  ;  3072  cu.  ft. 

6.  128  ft.  high. 

7.  6  cord  ft. 

8.  $2.95y\. 

9.  5ift. 

Arts.  432,  436, 

1.  63  perch  9  cu.  ft. 

2.  $300.96. 

3.  76545  bricks. 

4.  146966.4  bricks. 

5.  $1138.9896. 

3.  19 1  board  ft. 

4.  17^  board  ft. 

5.  110  board  ft. 

6.  $66. 

7.  20  board  ft. 

8.  120  board  ft. 

9.  42 1  b.  ft.  ;  3f  cu.  ft. 

10.  $5  50. 

11.  12  ft. 

12.  260  cu.  ft. 

13.  2  0.  4  ft. 

14.  500  ft. 

15.  $11.25. 

16.  65 1  ft. 

17.  $3>0125. 

18.  273;^  cu.  ft. 
$297.97,4,  cost. 


19.  $7,875. 

20.  3038  cu.  ft. 

Arts,  438,  439. 

3.  1795ff. 

4.  136*11  hhd. 

5.  21371 II  hbd. 

6.  6  ft.  11|1|  in. 

7.  267i|  ft. 

8.  $162.72  +  . 

9.  373|ff  cu.  ft. 

11.  208  bu. 

12.  7  ft.  9|  in. 

13.  5  ft. 

14.  10  ft. 

15.  $360. 

16.  44.8  bu. 

17.  16if  T. 

18.  12.6  T. 

Art.  441. 

1.  110.4  A. 

2.  $453.03125. 

3.  $708  and  270,  rem 

4.  8100  cu.  in. 

5.  9288  cu.  in. 

6.  2160  cu.  ft. 

7.  3f  cords. 

8.  31 1  If  cords. 

9.  $5.00. 

10.  $250. 

11.  $6. 

12.  114yV  sq.  yd. 

13.  156  sq.  yd. 

14.  72  yd. 

15.  85  yd. 

16.  $33825. 

17.  $30750. 

18.  $49.095^f. 

19.  880000  times. 

20.  20  da.  20  hr. 

21.  18849  ,V;1  wk. 

22.  2700  bricks. 

23.  369062  in. 

24.  17400  shingles. 

25.  144  farms. 

26.  220320  bricks. 

27.  30  da.  10  hr. 

28.  76  yr.  37  da.  7  hr. 

46  min.  40  sec. 


Answers, 


391 


Art.  452. 

13. 

8if%. 

17. 

$4800,  entire  cost. 

3. 

.40  or  40%. 

14. 

212|f%. 

$5.05 i%,  cost  per  bar 

4. 

.171  or  17|%. 

15. 

3|%. 

18. 

$16000,  whole  cost 

5. 

1*                     y  /*^ 

.30  or  30 /o. 

16. 

1%. 

$20  per  bar. 

6. 

.48  or  48^. 

17. 

18. 

16|%. 

Art,  47 S. 

Art.  460, 

19. 

66|%. 

1. 

$1347.80. 

o 

20. 

$2863  for  3d.              | 

2. 

$5.79. 

3. 
4 

5^1207. 

£2024.16. 

6506  bu.  3  pk.  2  qt. 

25342  5  9' 

1st. 

3. 

$537.50. 

5. 

45|Mt%. 

2d. 

4. 

$56.25. 

6. 

2240  lb. 

299%%, 

3d. 

5. 

$252. 

7. 

$62.50. 

6. 

$448,121. 

8. 

$588. 

Art,  466,           1 

7. 

$380.70." 

9. 
10. 
11. 

dE1218. 
$359.25. 
194625  A. 

3. 

4. 

288. 
2340. 

8. 
9. 

$1485. 

$1787.50. 

12. 

$2048.50. 

5. 

£3428|. 

10. 

$791,464. 

13. 

2725  ft.  3|  in. 

6. 

1000. 

11. 

$2843.75. 

14. 

$188058.33i. 

7. 
8. 

8(i00  yd. 
312. 

12. 

$312.0605. 

15. 

$432,  the  first. 

9. 

250. 

13. 

$1163.75. 

16. 

$5580,  the  second. 

10. 

$120. 

14. 

$29250. 

17. 

$7245. 

11. 

$40000. 

18. 

$3833.60. 

12. 

$21150. 

Art,  474, 

19. 

$911.25. 

13. 

62|. 

15. 

m%. 

14. 

46.8. 

16. 

22%  fo. 

Art.  462, 

15. 

43100. 

17. 

20%. 

16. 

$7200. 

18. 

31A%. 
30%. 

23. 

23. 

$6480. 
3724  HI. 

17. 

$184. 

19. 

24. 

$303000. 

.$1060. 

20. 

100%. 

25. 

10862  men. 

18. 

$3675. 

21. 

25%. 

26. 

1780  sheep. 

22. 

4HMf%- 

27. 

$10680. 

Art,  469, 

23. 

4f%. 

3. 

4856xV 

24. 

33i%. 

Art,  464, 

4. 

2281^3. 

25. 

23|i%. 

3. 

4. 

29i%. 
1%. 

5. 
6. 

26000. 
2200. 

Art.  475. 

5. 

8H%. 

rr 
1  . 

$3100. 

26. 

$5478.26^. 

6. 

6A%. 

8. 

S7000. 

■27. 

$17036.363^. 

7. 

11%. 

9. 

5200. 

28. 

$24375. 

8. 

63i%. 

10. 

2234:^. 

29. 

$4175.36. 

9. 

90  % ,  Henry. 

11. 

1363,V 

30. 

$6970. 

94%,  sister. 

12. 

2705  bu. 

3  pk.  4  qt. 

31. 

$371. 16|. 

10. 

m%. 

Apt. 

32. 

$516.25. 

11. 

156|  bu.  sold. 

13. 

$7000. 

33. 

$980.20. 

62^%. 

14. 

15000. 

34. 

$1634.71|. 

12. 

25  9;,  wife. 

15. 

$7179.18 

_'-". 

35. 

$3435  20. 

$3125,  each  child. 

16. 

$3085.71 

36. 

$1696.2281^. 

392 


Answers. 


Art.  47 <h 

39.  $77.23  fV. 

40.  loii.oaiif. 

41.  $1555.551. 

42.  $3235.29i-V. 

Art,  4S3. 

1.  $150.06. 

2.  $108,971-. 

3.  $46.41.  " 

4.  $109.37i. 

5.  $373.40:Cora. 
$7094.60,  p'd  own'r. 

C.  $486.87,  bill. 

$11995.13,  net  pro. 

7.  $2400. 

8.  .$8400,  amt.  of  sales. 

9.  $1750,  amt.  of  sales. 

10.  $1 8000, aTiit.of sales. 

11.  $27000,  selling-  j,r. 
$26595,  net  pro. 

12.  $2000.70. 

Art.  484, 

14.  $236.91. 

15.  $15506.23. 

Art.  485, 

17.  $1583.512/y. 

18.  $4345.36ff. 

19.  $4696.65^0%. 

20.  3200.38  bbl. 

21.  $49261. 08  o^.\. 

22.  500  robes. 

23.  $10024.45. 


Art.  500, 

$360. 
$112.50. 

$487.50. 

n%. 

21%. 
50%. 
$10520. 
$13600. 

Art.  501. 


10.  $16964.28). 


11.  $361. 538  rV 

12.  $6666.66|. 

Art,  505. 

13.  $202.50. 

14.  $202.50,  an.  prem. 
$5250,  amt.  in  20  yr. 

15.  Equal. 

Art.  515, 

2.  $62,548. 

3.  179.59. 

4.  $128,408. 

5.  .083,  rate. 

$441.75. 

Art.  522, 

2.  $5062.50. 

3.  $20475. 

4.  $1234.80. 

6.  $1320. 

7.  $7,095. 

8.  $492.1875. 

Art.  5,j6. 

3.  $551124,  int. 
$395.31,  amt. 

Art.  537. 

4.  $72.96,  int. 

5.  $1177.875,  amt. 

6.  $1103.52,  amt. 

7.  $10.05,  int. 

8.  $183.98625. 

9.  $1142.52,  amt. 

10.  $340.27|. 

11.  $51,548  +  . 

12.  $30.77. 

13.  $102.50. 

14.  $378,102. 

15.  $33.0338,  int. 

16.  $645.83,  amt. 

17.  $62.80,  amt. 

18.  $4.80. 

19.  $750.40,  amt. 

20.  $470.52,  amt. 

21.  $65.16~,  int. 

22.  $4662.25,  amt. 

23.  $134.78. 


24.  $20,326. 

25.  $232.73  +  . 

26.  $9808.81,  amt. 

27.  $17186.90,  amt. 

28.  $11578  53  +  ,  ami 

29.  S14472.096,  amt. 

30.  $55237.86,  amt. 

31.  $30,724. 


9.9 


$24.08. 


33.  $2741.15,  amt. 


Art.  539. 

2.  $381,277. 

3.  $1,343,  int. 

4.  $7,689,  int. 

5.  $45696. 

6.  $168,901. 

7.  $70,698,  int. 

8.  $1,759,  int. 

9.  $66,832,  int. 

10.  $5,684,  int. 

11.  $691,071,  amt. 

12.  $2879.854,  amt. 

13.  $4423.372,  amt. 

14.  $96.39,  int. 

15.  $340,277,  int. 


Art.  540, 

3.  $4,725. 

4.  $1.35. 

5.  $3,483. 

6.  $10.85. 

7.  $5.25,  int. 

8.  $6,431,  int. 

9.  $10,  int. 
10.  $31,  int. 

Art.  542. 

2.  $39,259. 

3.  $11,507. 

Art.  553. 

2.  $426.42. 

3.  $366,654. 

4.  $672,051. 

Art.  554. 

6.  $149,211. 

7.  $518,501. 


Ansivers. 


393 


Art,  55ij. 

2.  m%' 

3.  1',%. 

4.  6]-%. 

5.  5%. 

6.  9%. 

7.  ^fo. 

8.  6%. 

9.  7i%. 


Arts,  5iS(>f  5ii7, 

2.  1  yr.  10  mo.  28  d. 

3.  16  yr.  8  mo. 

4.  10%  10  yr. 

2.  $7142.857. 

3.  $11666.06|. 

4.  $12800. 

5.  $4237.288. 

6.  $446,428. 

7.  $1168.831. 


Art,  55f> 

2.  $102.04. 

3.  $139.50. 

4.  $209.02. 

5.  $348.21. 

6.  $1289.01. 

Art,  301, 

2.  $4690.34. 

3.  $560.36. 

4.  ^261.69. 

5.  $1524.46. 

6.  $1174.42. 

7.  $1194.05 

8.  $1520.12. 

9.  $1938.35. 


Ai't,  565, 

2.  $780,045,  pr.  worth. 
$70,205,  true  dis. 

3.  $1170.11.  pr.  vvortli. 
$102.39,  true  dis. 

4.  $2631.82,  pr.  worth. 
$263.18,  true  dis. 

5.  $4881.86,  pr.  worth. 
$768.89,  true  dis. 


6.  $9527.44.  pr.  worth. 
$472.56,  true  dis. 

7.  $41.60. 

8.  $3214|,  pr.  worth. 
$1607^33,  true  dis. 

9.  $12380.95. 

Art,  57 (K 

2.  $639,925. 

3.  $816.34. 

4.  $1258.84. 

5.  $18.61. 

6.  $821.76. 

Art,  571. 

2.  $518.45. 

3.  $4473.01. 

4.  $5342.81. 

Art,  574, 

2.  $418.50. 

3.  $511.65. 

4.  $657.90. 

6.  $2211.84. 

7.  $3147.54. 

8.  $3623.29. 

9.  $5606.25. 

10.  $8881.50. 

11.  $696.62. 

Art,  575, 

2.  $73.68. 

3.  $4.80. 

4.  $150. 

5.  $375. 

Art,  5 S3, 

2.  4  mo. 

3.  6  mo. 

4.  Hyr. 

5.  3  mo. 

6.  6|  mo. 

7.  2  yr.  3  mo. 

Art,  584. 

9.  Au^.  15th,  1879. 

10.  MaV  4th,  1880. 

11.  Aug.  19th. 


12.  Sept.  6th. 

13.  Sept.  21st. 

14.  Jan.  5th,  1881. 


Art,  5S(i, 


2.  Sept, 

3.  Nov. 
Bal. 

4.  Apr. 
Bal. 

5.  Dec. 
Bal. 

0.  July 
Bal. 

7.  Aug. 
Bal. 


,  2d. 

28th. 
due,  $150. 

1st. 

due,  $1730. 
20th,  1879. 
due,  $140. 
28th. 
due,  $100. 

13th. 
due,  $1275. 


Art.  601. 

1.  $280. 

2.  $750. 

3.  $510. 

4.  §420. 

Art,  602. 

5.  $2461. 
0.  $2604. 

7.  $10048.50. 

8.  $13725. 

9.  $12525. 

Art,  604, 

2.  15%. 

3.  m%. 
5.  m-fc. 

Art.  605. 

7.  $9750. 

8.  $10890. 

9.  $21875. 
10.  $47062.50. 

Art.  606. 

12.  125  shares. 

13.  50  shares. 

14.  60  shares. 

15.  73  shares. 

16.  40  shares. 


394 


Answers. 


Arfs.607,10. 

18.  66|%. 

19.  G2L%. 

20.  1331  fc. 

22.  $78750. 

23.  $14437.50. 

24.  $24040. 

25.  $28333.331. 
2C.  $6360. 

29.      4:fjfo. 

30.  5Hf%. 

31.  $666. 66|. 

34.  311^. 

35.  lUf%. 

Arts. 624,25. 


3. 

4. 

5. 

7. 

8. 

9. 
11. 
12. 
13. 
14. 
15. 
16. 


$862.75. 

$970,125. 

$2035. 

$5285.29. 

.$3445.75. 

$5075. 

$2439.02. 

$3419.69. 

$454.78. 

$2308.64. 

$4160.16. 

$2971.77. 


Arts, 632, 34. 

2.  $1683.22. 

3.  .$1843.092. 

5.  $145.54. 

6.  $195.26. 
8.  $675.50. 


Art.  635. 

10.  £1337,  8s. 

11.W. 

11.  12360  fr. 

12.  7740  fr. 

13.  3200  marks. 

14.  85331  marks. 

Art.  641. 

2. 
3. 
4. 


$499.20. 

$47.50. 
$97.50. 
$24.50. 


6. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 
23. 
24. 
25. 
26. 
27. 

Oft 


$35.28. 
$37.50. 
$7.44. 
$5.15. 
756  m. 
5624  bu. 
5106  m. 
84 II  Km. 
£1." 

$53333.33. 
$427.50. 
$75.74. 
$23,125. 
60  d. 
50  d. 
4:1  d. 
1  ^\  mill. 
2y\  lir. 
8ds. 
4i  hr. 


Art.  643. 

2.  99  lb.  lOf  oz. 

3.  98  lb.  7/3  oz. 

4.  69T.  1285 fib. 

5.  .07AV 

6.  93bu.5.12qt. 

7.  $0,831. 

8.  818.568  m. 

9.  4858  55+  lb. 

10.  9581b. lO.loz. 

11.  $6.75. 

12.  189  yd. 

13.  10|doz. 

14.  325  lb. 

Art.  645. 

2.  $12. 

3.  $80. 

4.  $250. 

5.  $250. 

6.  $872.50. 

Art.  647. 

2.  107  b.  44  q,  A. 
85  1).  22 f  q.,  B. 
87  b.  4iq.,('. 


3.  $600,  A's. 
$375,  B's. 
$525,  C's. 

4.  666|bar.,A's. 
800  bar.,  B's. 
1000  bar.,  C"s. 
5331  bar.,D's. 

5.  $315,  A's. 
$525,  B's. 
$420.  C's. 

6.  $1250,  X's. 
$1750,  Y's. 
$2000,  Z's. 

7.  $0.66|. 
$200,  1st. 
$266.66|,  2d. 
$333,331,  3d. 

8.  $0.80. 

9.  $64.14,  A's. 
$105.12,  B's. 
$147.76,  C's. 

10.  $0.10  on  $1. 
$500,  B  rec'd. 

11.  $0.4223  +  . 

12.  100  bar.,  A's. 
66|  bar.,  B's. 
33^  bar.,  C's. 

14.  $100,  A. 
$120,BandC. 

15.  $30. 

16.  S40.02,  A's. 
$88.28,  B's. 
$117.70,  C's. 

17.  $332.50.  S's 
$525,  Jones'. 

18.  $508.83,  A's. 
$677.75,  B's. 
$938.42,  C's. 

Art.  649. 

2.  8581. 


3. 
4. 
5. 
6. 

7. 

8. 

9. 
10. 
11. 
12. 


2554^. 
2666f.' 

$288. 

$1.20. 
875  pears. 
425. 
9S0. 
270  mi. 
$56. 


13.  385. 

14.  $2.10. 

15.  22  T.  15001b. 

16.  2  coats. 

17.  21  tubs. 

19.  240slieep. 

20.  $288. 

21.  1440  men. 

22.  48  ft. 

23.  $296. 

24.  $14400. 

25.  72  yr. 

26.  21i9i. 

27.  72  pupils. 

28.  $15600. 

29.  60  trees. 


Art.  650. 


3. 

4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 


40  shares, 

$62.30. 

$4932.64. 

$2187.50. 

$16841. 

8%. 

$3869.72  +  . 

$1619.31. 

$1010.50. 


12.  Um. 


13. 


14. 
15. 


$900,  A's. 
$1040,  B's. 
$1060,  C's. 

$770,625. 
$3894.4325. 


Art.  651. 

2.  324  in  one. 
432  in  other. 

3.  312  in  one. 
936  in  other. 

4.  108|,  one. 
326 1,  other. 

5.  151  A.,  one. 
604  A.,  other. 

6.  $1312.89,one. 
$1641.11,ot'r. 

7.  180,  first, 
240,  second. 
300,  third. 

9.  378,  1st. 
252.  2d. 
315,  3d. 
11.  205 


Ansivers, 


395 


12. 
13. 
14. 

15. 


11. 

1211  Kl. 
40  peaches. 
80  pears. 
160  apples. 
64,  1st. 
32,  2d. 
96,  3d. 


Art,  652, 

17.  7i  ds. 

18.  2|  mo. 

19.  20  men. 

20.  720  mi. 

21.  224  bu. 

Art.  653, 

23.  90  cts.,  1  pt. 
40  cts.,  14  pt. 

24.  30  cts.,  I'pt. 
24  cts.,  5  pt. 

Art,  668, 

1.  3|. 

2.  2/j. 

3.  214. 
4  2  3 

K       13 
»•  3^- 

6. 

7. 

8. 

9. 
10. 
11. 
12. 


1 1' 

27 

6?- 
11 
2F' 
8 

7- 

4 

9- 

_8_ 

15- 

49 

73' 

13:  16. 

14.  11. 

15.  1151. 


16. 

16|. 

17. 

31 

18. 

ih- 

19. 

32. 

20. 

3\- 

J 

Art. 

21. 

7. 

22. 

A- 

670. 


23.  f 

24.  112. 

25.  f|. 

26.  VW- 

27.  192. 

28.  432. 

29.  I183. 

2.  70. 

3.  2550. 

4.  7xV. 

5.  48. 

6.  288. 
7.'  375. 

8.  27.3. 

9.  35  vests. 

10.  736.?  lb. 

11.  131  gal. 

12.  45. 

13.  75. 


Art,  687, 

3.  3|d. 

4.  $515. 

5.  £22,  10s. 

6.  1440  min, 

7.  $3.75. 

8.  $5000. 

9.  13^  mo. 

10.  $60. 

11.  75  ft. 

12.  lOOd. 

13.  105  1iektars. 


Art,  692, 

3.  $3.44. 

4.  $5.01. 

5.  108.5  Kil. 

6.  69f«3Hl. 

7.  318|Km. 

8.  133  3^  spoons. 

9.  $251|. 

10.  362  d. 

11.  £51,  3s.  2d. 

12.  £3,  12s.  6d. 


13.  £41,  12s. 

14.  3f  hr. 

15.  5  min. 

16.  12  br. 
$195.14. 
U\9,  in. 
164L  yd. 


6d. 


17. 

18. 
19. 


Art.  605, 


3. 

4. 
5. 
6. 

7. 

8. 

9. 
10. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 


768  m. 
96  men. 
10  men. 
6d. 
7.2  d. 
170|  bu. 
80  d. 
6  men. 
$18. 
90d. 
25  lb. 
60  men. 
18  d. 
3s.  1.7d. 
792  pr. 


Art.  608, 

3.  72,  1st. 
96.  2d. 
144,  3d. 

3.  22  sb.,  1st. 
66  sb.,  2d. 
110  sb.,  3d. 

4.  120  bu.  oats. 
160bu.  wlie't. 
220  bu.  corn. 

5.  $0.88,  pears. 
$1.76,or'ng's. 
$2.64,merns. 

6.  $497,  1st. 
$745.50,  2d. 
$994.  3d. 
$1242.50,4tli. 


Art.  706, 

2.  t,  A's. 
h  B's. 

3.  $120.  A's. 
$160,  B's. 
$200,  C's. 


4.  $342.86,  A's. 
$457.14,  B's. 

$685.71,  ("'.s. 
$914.29,  D's. 

Art,  708. 

6.  $50,  A's. 
$100,  B's. 

7.  $100,  A's. 
$120,  B's. 
$120,  C's. 

8.  $30. 

9.  $40.02,  A's. 
f88.28,  B's. 
$177.70,  C's. 

10.  $600,  A's. 

11.  $589.47,  A's. 
$1129  82,  B's. 
$1670.18,  C's. 
$2210  53,  D's. 

12.  $5090. 

13.  16|%. 

14.  10%,  or 
$1500,  A's 

loss. 

Art,  718, 

2.  216. 

3.  729. 

4.  53824. 

5.  3125. 

6.  2401. 

7.  42875. 

8.  65.450827. 

9.  8.003600540- 

027. 

10.  27013502.25- 

0125. 

11         **!- 

13.  111.56640625. 


Art,  710, 

4.  900  +  240 

+  16. 

Art,  734, 

4.  24. 

5.  40. 

6.  35. 


396 


Answers. 


9. 
10. 

11. 

12. 
13. 
14. 
15. 
16. 
17. 
18. 


17.07  +  . 
29.4  +  . 
23.20  +  . 
7.859  +  . 
92.06  +  . 
1110.016  +  , 
2305.317 +  , 
16.96  +  . 
.881  +  . 
32.768. 
.0331. 
785.64 


Art,  735, 

3.  i. 

4.  I 

5.  ^. 

6.  .36288  +  . 

7.  1. 

8.  .4848  +  . 

10.  ^. 

11.  6|. 

12.  ri 

13.  10|. 

14.  9|. 

15.  .751  +  . 

16.  14A. 

17.  1.7320508. 

18.  3.46410161 +  . 

1.  45  yd. 

2.  952. 

3.  783.836  rd. 

4.  72  ft. 

5.  24  rds. 

6.  390  rds. 

7.  1395  yd.  sq. 


ArU  736, 

9.  8. 

10.  15. 

11.  30. 

12.  56. 

13.  72. 

14.  38.8844442. 

15.  65.72  +  . 

16.  .08. 

17.  2. 

18.  .18. 

19.  1. 


20.  A. 

21.  M. 

22.  .3. 

OQ  3 1 


24.  80  rd.,  width. 
160  rd.,  length. 

25.  80  rd.,  breadth. 
320  rd.,  length. 

Art,  74,3. 

27.  10  yd. 

28.  50  m. 

29.  200  m. 


Art.  744, 

31.  60  ft. 

32.  24.98  ft. 

33.  8  yd. 

Art.  745. 

35.  24  ft. 

36.  103.614+  ft. 

37.  42.4264  rd. 

38.  40  rd.,  length,  side. 
56.5685  rd.,  diago- 
nal. 

39.  50  ft.,  floor   diago- 

nal. 
51.92+     ft.,    other 
diagonal . 

40.  65.802+  rd. 

41.  56.5685  ft. 

42.  75.816+  ft. 

Art.  747. 

3.  15  min. 

4.  18  in. 

5.  30  in. 

6.  24.49+  vd. 


1. 
o 

3. 
4. 
5. 
6. 


Art.  750, 

2  fig 
2  fig 

2  fig 

3  fig 
3  fig 
2fiff 


Art,  752, 

2.  303  +  3(30-^x2) 

+  3(30x22) +  2'. 

Art,  754. 

4.  24. 

5.  83. 

6.  72. 

7.  125. 

8.  103. 

9.  1331. 

10.  3002. 

11.  2.3. 

12.  4.5. 

13.  .632  +  . 

14.  5.48. 

15.  49.68. 

Art,  755, 

17.  .601  + 

18.  i\. 

19.  f. 

20.  If. 

21.  2.39  +  . 


00, 


31. 


23.  1.2599 +  . 

24.  1.442249 +  . 

Art,  756. 

1.  73  in. 

2.  364  ft. 

3.  108  yd. 

4.  8  ft. 

5.  8  ft.  6.44  in. 

6.  9  ft.  5.3  in. 

7.  58.8+  ft. 

Art.  75S, 

3.  137.48+  lb. 

4.  $121362.96. 

5.  1562.5  cu.  ft. 

6.  23^V  T. 

7.  163840000  1b. 

8.  llll|hhd. 

Art.  759. 

10.  7.2+  ft. 

11.  8  in. 

12.  3  ft. 

13.  4  ft. 


> 


Answers. 


397: 


Art,  76S. 


1.  15. 

Art,  769, 

2.  16. 

Art.  770. 

3.  38. 

4.  10. 

5.  27. 

6.  $550. 

Art.  771. 

2.  8  children. 

3.  23. 

4.  381  d- 

Art.  77*^. 

2.  3  yr. 

3.  $3. 

4.  ^%. 

5.  ^V%. 

^1  ^  773. 

2.  78  strokes. 

3.  180000. 

4.  $651. 

3.  $455.81. 

4.  $1605.87,  anit. 
$32,76.975,  amt. 


Art.  7S0. 


2    -^-^- 

'*•      243- 


Art.  7 

2.  1456. 

3.  $255. 

4.  $40.95. 

5.  $5314.40. 


SI. 


Art.  S09, 

2.  450  sq.  ft. 

3.  4914  sq.  ft. 

4.  46  A.  17i  sq.  rds. 

5.  4556]  sq:  yd. 

6.  250  sq.  ft.* 


Art.  810. 

2.  50.91  +  sq.  yd. 

3.  198.43+  sq.  ft. 

4.  7  A.  58.14  sq.  rd. 

Art.  811. 

2.  3  yd. 

3.  150  rd. 

Art.  812. 

2.  80  rd. 

3.  52.6  yd. 

Art.  821. 

3.  67iA. 

4.  640  A. 

5.  26  A.  65  sq.  rd. 

6.  50  rd. 

7.  80  rd. 

9.  4  A.  75  sq.  rd. 

Art.  822. 

2.  368sq.  yd. 

3.  558  sq.  rd. 

Art.  823. 

2.  50  A.  125  sq.  rd. 

3.  12480  sq.  yd. 

Art.  829. 

2.  141.372  yd. 

3.  314.16  rd. 

Art.  830. 

2.  30  rd. 

3.  200  yd. 

4.  8  rd. 

5.  16  ft. 


9. 
10. 


Art.  831. 

7854  sq.  ft. 
11309.76  sq.  rd. 
2037.178+  sq.  yd. 
15.91  ft. 
10.472  ft. 
706.86  sq.  ft. 
203.7178  A. 
7.97  rd. 


Art.  841. 

3.  126  sq.  ft. 

4.  54  sq.  in. 

5.  152  sq.  ft. 

6.  576  sq.  ft. 

7.  640  sq.  ft. 

8.  4084.08  sq.  ft. 

Art.  842. 

3.  375  cu.  ft. 

4.  9200  cu.  ft. 

5.  565.488  cu.  ft. 

6.  45945.9  cu.  ft. 

Art.  846. 

3.  7744  cu.  ft. 

4.  3817.044  cu.  ft. 

5.  441  cu.  ft. 

Art.  850. 

2.  4.91  sq.  ft. 

3.  201062400  sq.  m 

Art.  831. 

2.  523.6  cu.  in. 

3.  259,777,100,10S  cu 

miles. 

4.  381.7+  cu.  in. 

Art.  852. 

2.  mf.  ft. 

Art.  855. 

2.  44.982  gal. 

3.  1059.5286  liters. 

4.  548.4375  gal. 

Art.  856. 

1.  2526rVT. 

2.  378if  T. 

Art.  858. 

1.  164  girls. 
304  pupils. 

2.  62f4f  rd.,  breadth. 
330f  A. 

3.  .568431. 

4.  .881  A. 


398 


Answers. 


5.  45f  ft.,  height 

6.  $1,462. 

7.  8157  da. 

8.  $28.95. 

9.  $4702.50. 

10.  7. 

11.  5  bbl.  152  lb. 

12.  88704  steps. 

13.  $64G0.40. 

14.  12  ft. 

15.  $34.57. 

16.  39.163  yd. 

17.  3949,  one. 
4705,  other. 

18.  $462.50,  one. 
$1037  50,  other. 

19.  469,  less. 
1407,  greater. 

20.  $315,  B's  share. 
$1260,  A's  share. 

21.  511?,  one. 
m~ij,  other. 

22.  1  =  g-.  c.  d. 
261648  =  1.  c.  111. 

23.  23  p.  of  36  yd. 

24.  16  A. 

25.  $1258250. 

26.  139180. 

27.  132x7x2. 

28.  4  days. 

29.  15,  j;-.c.  <l. 
77805,  l.c.m. 

80.  72,  2d. 
144,  4th. 
28,  7,  60  and  455 

^^-  105  • 

32.  $720. 

33.  .0322465. 

34.  30  da. 

35.  $611,625. 

36.  $10.26. 

37    91  lb.  6.5  oz. 

38.  $1914.28|. 

39.  $31232.  ' 

40.  74f  ft. 

41.  6,  l.c.m. 

42.  171  da. 

43.  $43.52. 

44.  $2.55. 

45.  8579991 3. 

46.  $17. 


47.  $1260. 

48.  $.04. 

49.  $21.67  lost. 

50.  .15;-^. 

51.  $86.40,  or  $90. 

53.  65  mi. 

54.  24  men. 

55.  350. 

56.  $18.78. 

57.  24. 

58.  56.568  mi. 

59.  48r.l-  yd.,  or  50  yd. 

60.  2l6'yr.  97da.  21  hr. 

38  min.  15  sec. 

61.  $941.14. 

62.  $48.92. 

63.  $60. 

64.  $44.58. 

"'•      50336' 

68.  $1700. 

69.  7695. 

70.  A.  e.  111.  of  the  Nos. 

71.  7  hr.  19i\  min. 

24|f  min. 

70      .T 

''^'    Toff- 

73.  $3.35. 

74.  907.1  lb. 

75.  $18/25,  int. 
$260  71,  prin. 

77.  $44.26. 

78.  45,  longer. 
18,  shorter. 

79.  14  ;  21 ;  12  rows. 

80.  17i 

81.  $22.37. 

82.  $10.94. 

83.  33i%. 

84.  75>. 

85.  $201.60. 

86.  $1546.40. 

87.  20%. 

88.  f^,  A's  share. 
^Yw^,  B's  share. 
^Wt'  C's  share. 

89.  $1.76tV 

90.  7%. 

91.  $545.73. 

92.  14?,  da. 

93.  $2000. 

94.  $5.00 


95.  40  vd. 

96.  $185. 

97.  $9000. 

98.  $85. 

99.  250  shares. 

100.  $18277.91. 

101.  101%. 

102.  16326  yr.   11   mo 

4.44  da. 

103.  $3.41. 

10-1.   $1320,  As  share. 
$880,  B's  share. 
$440,  O's  share. 

105.  103.9. 

106.  80  rd. 

107.  SOg^  vd.,  or  52  yd. 

108.  13.34  + ,  diagonal. 
11250  lb. 

109.  $10.3125. 

110.  $53.90. 

111.  260  bu.  11.424  qt. 

112.  $550.92. 

113.  71f%. 

114.  $1525.55. 

115.  16875. 

116.  240  rd. 

117.  $408.96. 

118.  $175. 

119.  $362.30. 

120.  80.6  ft. 

121.  56"  8'. 

122.  1  da.  6  hr.  25  min. 

123.  4165  cu.  ft. 

124.  470 1  cu.  yd. 

125.  1190sq.  rd. 

126.  2101  mi. 

127.  107fV    lb.,    entire 

weight. 
35f|  lb.,  average 
weight 
128. 

129.  ^T 

130.  $10. 

131.  88°  45'. 

132.  169UCU.  ft. 

133.  $401.20. 

1.34.  22  min.  40  sec. 

135.  .635. 

136.  $24634.63,  inves't. 
$615  87,  com. 

137.  $18.75. 


8 

6^  qt. 


Ansvjers. 


399 


138.  $6. 

139.  6|lir. 

140.  34J4  T. 

141.  14.5  ft. 

142.  486  tiles. 

143.  47J/j%. 

144.  7()jU%- 

145.  $3.75. 

146.  3432  ft. 

147.  .44A. 

1 48      30  -•'  **^- 

X'±0.      .OW,  0  0  0- 

149.  3|mo. 

150.  /«. 

151.  d^jdn. 

152.  .299980317. 

153.  8.55  ft.,  nearly. 

154.  5  h.  56  m.  32  s. 

155.  $99. 

156.  POI.72. 

157.  $12.50. 

158.  38.47  sq.  rd. 

159.  42  in. 

160.  i\%,  one. 
-j^j;,  other. 

161.  $125000. 

162.  $7.22. 

163.  Iff. 

164.  1500  lb.,  Nit. 
250  lb.,  S.  and  C. 

165.  1.1071. 

166.  $1200,  IPt. 
$1600,  2d. 
$1800,  3d. 

167.  $190000. 

168.  $167.20. 

169.  93|%. 

170.  $107.10. 

171.  $9.70. 

172.  2  yr.  4^  mo. 

173.  $300.  ' 

174.  15fda. 

175.  $1000,  or  10%. 

176.  384  sq.  ft. 

177.  .062. 

178.  $654.50. 

179.  35    30'  eastward. 

180.  Iff. 

181.  525. 

182.  $413.44. 

183.  59.92  sq.ft. 


184.  249.41  sq.  yd. 

185.  $55924.05. 

186.  $130.80. 

187.  $48.05,  A's. 
.$69.95,  B's. 

188.  66.98  lb.  Troy. 

189.  40. 

190.  30.7125  A. 

191.  192  lots. 

192.  300  strokes. 

193.  2,  com.  difference. 
99,  sum. 

194.  15  holidays. 
$360. 

195.  70  A.  109.76  sq.  rd. 

196.  $3600.84. 

197.  6  bonds. 

198.  $284.06. 

199.  562.34  mi.,  nearly. 

200.  $19.40. 

201.  123.888  sq.  ft. 
7.87  ft. 

202.  2037.178  sq.  yd. 

203.  1558.75  cu.  ft. 

204.  540  en.  ft. 

205.  9200  ca.  ft. 

206.  13750  cu.  ft. 

207.  22619.52  cu.  ft. 

208.  20420.40  cu.  ft. 
209. 
210.  3048. 


211 

•  3tV%. 

212 

•    llyf 

Art,  859. 

1. 

63,  g-.  c.  d. 

2. 

1.4142. 

3. 

72  men. 

4. 

10%. 

5. 

3654.7  meters. 

6. 

3|- 

7. 

73677.6846  cu.  dm 

8. 

71ir. 

9. 

$2500. 

10. 

M.  .89,11- 

11. 

m- 

12. 

825 
1288- 

13.  .0245i|f. 

14.  .831114  A.,  or 

.831433+  A. 


15.  160  sq.  dm. 

ifi    -;  1361 

17.  .218. 

18.  .059. 

19.  $7295.43. 

20.  lli%. 

32.  4|  oz.,  nearly. 
58.293+  meters 

23.  41  pt. 

24.  7501b. 

25.  $2331.12//j,  A's 
$4662.24i|f,  B's 
$5006.61i|f,  C's. 

26.  4.38  +  . 

27.  360,  I.e.  111. 
2,  j» .  c.  d. 

28.  7.6199  meters. 

29.  1879. 

30.  $.88|. 

31.  $1142.86. 

32.  2.358^,  sum. 
.291f ,  product. 

33.  1566.712 +  . 

34.  1880.0001 +  . 

35.  488.2468  Kg. 

36.  36|f%. 

37.  1.178*. 

38.  12500  bricks. 

39.  $5061.68. 

40.  $21000  at  first. 
87if  %  loss. 

41.  $8400. 

42.  $7000. 

43.  $1210.59. 

44.  57.3332. 

45.  3.128. 

46.  3,  g.  c.  d. 

47.  504,  I.e.  111. 

48.  63^07.       • 

49.  9  o'clock  54  m.  23 3  & 

50.  .059375  day. 

51.  If 

52.  $'301.11. 

53.  4%  loss. 

54.  2.37. 

55.  1.60933  Km. 

56.  ly^. 


400 


Ansivers. 


57. 

.0096048. 

Art,  SS4, 

58. 

4  dm.  6  cm.  3  m. 

1. 

.8. 

59. 

1.05  books. 

60. 

37|  cts. 

3. 
3. 

.875. 
.75. 

61. 

$1060. 

4. 

.575. 

63. 

4.5  meters. 

5. 

.088. 

63. 

.000001. 

6. 

.8125. 

1.002001. 

7. 

.4375. 

64. 

3.331. 

8. 

.857143. 

65. 

2^  da. 

9. 

.3. 

66. 

$34.23. 

10. 

.761904. 

67. 

$3313.50. 

11. 

.36. 

68. 

$3447.50. 

69. 

14  f,. 

13. 

.17073. 

70. 

$3087.75. 

13. 

.416. 

71. 

13i  mo. 

14. 

.53. 

73. 

$9973.97.       ; 

15. 

.590. 

73. 

33.419. 

16. 

.36. 

74. 

7:\0. 

17. 

.313. 

75. 

$13530  34. 

76. 

1.60933  Km. 

18. 

.31. 

77. 

.5. 

19. 

.484375. 

78. 

1  yr.  7  mo.  6  da. 

30. 

.13. 

79. 

8%. 

80. 

735.6  liters. 

Art,  885. 

81. 

-iz- 

3. 

2 

83. 

8000  turns,  nearly. 

o 

11' 

83. 

$.95,  difference. 

o. 

TT- 

84. 

$381. 

4. 

23 
33- 

85. 

85018983. 

5. 

_41_ 
3  3  3* 

86. 

9tV. 

6. 

il 
37* 

87. 

500.003,  sum. 

7. 

6 

TTT* 

499  998,  difference. 

8. 

97 

88. 
90. 

5  Kg.  317.4  cr. 
13.573  +  . 

9. 
10. 

65 
4  2 

91. 

3.9  hektoliters. 

TOT- 

11. 
13. 

13. 

35121 

93. 
93. 

O  851 

\fo  premium. 

999¥¥' 

1 

94. 

33.713+  liters. 

1 

13* 

95. 

7^  cords. 

96. 

5  mo.  4  da. 

Art.  886, 

97. 

.495  +  . 

15. 

178 
2T5- 

98. 

8.65  Km. 

16. 

11_3 
TT25- 

99. 

31  bu.  3  pk.  5  qt. 

17. 

2933 
7¥Fff- 

18. 

19. 
30. 
21. 
33. 

1 

.!¥» 

^7' 
361 

^so- 
ls 
T¥- 

127457 
1560I)OOTy* 

Art, 

887, 

33. 

1.0897. 

34. 

.3377. 

Art, 

895, 

1. 

103f  A. 

3. 

$138.57 

3. 

$533.33 

4. 

$160. 

5. 

1930  A. 

$43530 

gain. 

Art. 

905, 

3. 

$1836.66. 

Art, 

910, 

3. 

$0.09^^7. 

Art, 

911. 

4. 

3  lb.  at 

9  cts. 

2  lb.  at  11  cts. 

4  lb.  at  14  cts. 

5. 

3  lb.  at  15  cts. 

3  lb.  at  18  cts. 

1  lb.  at 

31  cts. 

4  lb.  at 

33  cts. 

Art, 

912, 

7. 

35    bu. 

at  40  cts 

10    bu. 

at  45  cts 

8i  bu. 

at  56  cts 

161  bu. 

at  65  cts 

J- 


Thomson,  J*Il 


Complete  ^;raded  arthinie-  Dept» 
tio,  oral  and  written, 


upon  the  mau 
or  instruct 


/<s 


ii' 


/ 


otive  method 
on — — 


-idoo. 


M 


fy^ 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


A  Hand-Book  of  Mythology : 

Myths  and  Legends  of  Ancient  Gkeece  and  Rome.     Illustrated 
from  Antique  Sculptures.     By  E.  M.  Berens.    330  pp.    16mo,  cloth. 

The  author  in  this  volume  gives  in  a  very  graphic  way  a  lifelike  pic- 
ture of  the  deities  of  classical  times  as  they  wei-e  conceived  and  worshiped 
l)y  the  ancients  themselves,  and  thereby  aims  to  awaken  in  the  minds  of 
young  students  a  desire  to  become  more  Intimately  acquainted  with  the 
noble  productions  of  classical  antiquity. 

In  the  legends  which  form  the  second  portion  of  the  work,  a  picture,  as 
it  were,  is  given  of  old  Greek  life;  its  customs,  its  superstitions,  and  its 
princely  hospitalities  at  greater  length  than  is  usual  in  works  of  the  kind. 

In  a  chapter  devoted  to  the  purpose,  some  interesting  particulars  have 
been  collected  respecting  the  public  worship  cf  the  ancient  Greeks  and 
ilomans,  to  which  is  subjoined  an  account  of  their  principal  festivals. 

The  greatest  care  has  been  taken  that  no  single  passage  should  occur 
throughout  the  work  which  could  possibly  otZend  the  most  scrupulous  deli- 
cacy, for  which  reason  it  may  safely  bo  placed  in  the  hands  of  the  young. 


RECOMMENDATIONG, 

*'  Pifty  years  aso  compeuds  of  mythology  were  as  common  as  they  were  useful, 
hut  of  late  the  youthful  student  has  been  relegated  to  the  classical  dictionary  for  the 
information  which  he  needs  at  every  step  of  his  pros;ress.  The  legends  and 
myths  of  Greece  and  Rome  are  interwoven  with  our  hterature,  and  the  general 
reader,  as  well  as  the  classical  student,  is  in  need  of  constant  assistance  to  enable 
him  to  appreciate  the  allusions  he  meets  with  on  almost  every  pace.  The  classical 
dictionary  is  not  always  at  hand,  nor  is  there  always  time  to  find  what  is  wanted 
amid  Its  full  details,  and  the  reader  is  thus  often  obliged  to  answer  "no"  to  the 
question,  "  Understandest  thou  what  thou  readest  ?  "  Tiiis  handbook,  by  Mr. 
JJerens,  is  intended  to  obviate  the  difficulty  and  to  supply  a  want.  It  is  compact, 
and  at  the  same  lime  complete,  and  makes  a  neat  volume  for  the  ftudy  table.  It; 
gives  an  account  of  the  Greek  and  Roman  Divinities,  both  Majores  and  Minores, 
of  their  worship  and  the  fesiivals  devoted  to  them,  and  closes  with  sixteen  classical 
legends,  beginning  with  Cadmus,  who  sowed  the  draron's  leeth  which  sprang  up 
into  armed  men,  and  ending  with  a  wifely  devotion  or"  Penelope  and  its  reward. 
The  volume  is  not  one  of  mere  dry  detail,  but  is  enlivened  with  pictures  of  classi- 
cal life,  and  its  illustrations  from  ancient  eculpturo  add  greatly  to  its  interest."— 
*'  The  Churchtnan,'"  New  YorTc  City. 

*'  The  importance  of  a  knowledge  of  the  myths  and  legends  of  ancient  Greece 
and  Rome  is  fully  recognized  by  all  classical  teachers  and  students,  and  also  by  the 
intelligent  general  reader  ;  for  our  poems,  novels,  and  even  our  daily  newspapers 
abound  in  classical  allusions  which  this  work  of  Mr.  Berens' fuliy  explains.  It 
is  appropriately  illustrated  from  antique  sculptures,  and  arranged  to  cover  the  first, 
eecond  and  third  dynasties,  the  Olympian  divinities,  Sea  Divinities,  Minor  and 
Roman  divinities.  It  also  explains  the  public  worship  of  the  ancient  Greeks  and 
Romans,  the  Greek  and  Roman  festivals.  Tart  II.  is  devoted  to  the  legends  of  the 
ancients,  with  illustrations.  Every  page  of  this  book  is  interesting  and  instruc- 
tive, and  will  be  found  a  valualile  introduction  to  the  study  of  classic  authors  and 
assist  materially  the  labors  of  both  teachers  and  students.  It  is  well  arranged  and 
wisely  condensed  in'o  a  convenient-sized  book,  ]-2mn,  330  p;iges,  beautifully 
printed  and  tastefully  bound."—"  Journal  of  Education,''''  B(t>ton,  3Ja$s. 

"It  is  an  admirable  work  for  students  who  desire  to  find  in  printed  form  the 
facts  of  classic  mythology."— .ffcv.  L.  Clark  Seelye,  Fres.  Smith  College,  Northamp- 
ton, Mass. 

"  The  subject  is  a  difficult  one  from  the  nature  and  extent  of  the  materials  and 
the  requirements  of  our  schools,  llie  author  avoids  extrefaie  theories  and  states 
clearly  the  facts  with  modest  limits  of  interpretation.  I  think  the  book  will  take 
well  and  wear  Avell."— C.  F.  F.  Bancroft,  Fh.D.,  Frin.  Fhillips  Academij,  Andover, 

'^"^^'  Price,  by  Mail,  Post-paid,  $1.00. 

Clark  k  Maynard,  Publishers,  New  York. 


Yb  Joa^^ 


Two-Book   Series  of  Arithmetics. 

By  James  B.  Thomson,  LL.D.,  author  of  a  Matliematical  Course. 

1.  FIRST    LESSONS   IN   ARITHMETIC,   Oral    and    Written. 

Fully  and  handsomely  illustrated.    For  Primary  Schools.    144  pp. 
16mo,  cloth. 

2.  A  COMPLETE  GRADED  ARITHMETIC,  Oral  and  Writ- 

ten, upon  the  Inductive  Method  of  Instruction.     For  Schools 
and  Academies.     400  pp.     12mo,  cloth. 

This  entirely  new  series  of  Arithmetics  by  Dr.  Thomson  has  been 
prepared  to  meet  the  demand  for  a  complete  course  in  two  books.  The 
following  embrace  some  of  the  characteristic  features  of  the  books : 

First  Lessons.— This  volume  is  intended  for  Primary  Classes.  It  is 
divided  into  Six  Sections,  and  each  Section  into  Twenty  Lessons.  These 
Sections  cover  the  ground  generally  required  in  large  cities  for  promotion 
from  grade  to  grade. 

The  book  is  handsomely  illustrated.  Oral  and  slate  exercises  are  com- 
bined throughout.  Addition  and  Subtraction  are  taught  in  connection, 
and  also  Multiplication  and  Division.  This  is  believed  to  be  in  accordance 
with  the  best  methods  of  teaching  these  subjects. 


Complete  Graded.— This  book  unites  in  one  volume  Oral  and 
Written  Arithmetic  upon  the  inductive  method  of  instruction.  Its  aim  is 
twofold :  to  develop  the  intellect  of  the  pupil,  and  to  prepare  him  for  the 
actual  business  of  life.  In  securing  these  objects,  it  takes  the  most  direct 
road  to  a  practical  knowledge  of  Arithmetic. 

The  pupil  is  led  by  a  few  simple,  appropriate  examples  to  infer  for 
himself  the  general  principles  upon  which  the  operations  and  rules  depend, 
instead  of  taking  them  upon  the  authority  of  the  author  without  explana- 
tion. He  is  thus  taught  to  put  the  steps  of  particular  solutions  into  a 
concise  statement,  or  general  formula.  This  method  of  developing  prin- 
ciples is  an  important  feature. 

It  has  been  a  cardinal  point  to  make  the  explanations  simple,  the  steps 
in  the  reasoning  short  and  logical,  and  the  definitions  and  rules  brief,  clear 
and  comprehensive. 

The  discussion  of  topics  which  belong  exclusively  to  the  higher  depart- 
ments of  the  science  is  avoided ;  while  subjects  deemed  too  diiiicult  to  be 
appreciated  by  beginners,  but  important  for  them  when  more  advanced, 
are  placed  in  the  Appendix,  to  be  used  at  the  discretion  of  the  teacher. 

Arithmetical  puzzles  and  paradoxes,  and  problems  relating  to  subjects 
having  a  demoralizing  tendency,  as  gambling,  etc.,  are  excluded.  All  thjit 
Is  obsolete  in  the  former  Tables  of  Weights  and  Measures  is  eliminated,  and 
the  part  retained  is  cori-ected  in  accordance  with  present  law  and  usage. 

Examples  for  Practice,  Problems  for  Review,  and  Test  Questions  are 
abundant  in  number  and  variety,  and  all  are  different  from  those  in  the 
author's  Practical  Arithmetic. 

The  arrangement  of  subjects  is  systematic;  no  principle  is  anticipated, 
or  used  in  the  explanation  of  another,  until  it  has  itself  been  explained. 
Subjects  intimately  connected  are  grouped  together  in  the  order  of  their 
dependence. 

Teachers  and  School  OflBcers,  who  are  dissatisfied  with  the  Arith- 
metics they  have  in  use,  are  invited  to  confer  with  the  publishers. 


Clark  &  MAYNARD,  Publishers,  New  York. 


